Chapter VII: Book IV (3)

It is to be noted that the figures appear reversed in the perspective. That is, in the geometrical triangle the base at _ab_ is uppermost, whereas in the perspective _ab_ is lowermost, yet both are nearest to the ground line.

LVI

HOW TO PUT A GIVEN SQUARE INTO ANGULAR PERSPECTIVE

Let _ABCD_ (Fig. 115) be the given square on the geometrical plane, where we can place it as near or as far from the base and at any angle that we wish. We then proceed to find its perspective on the picture by finding the perspective of the four points _ABCD_ as already shown. Note that the two sides of the perspective square _dc_ and _ab_ being produced, meet at point _V_ on the horizon, which is their vanishing point, but to find the point on the horizon where sides _bc_ and _ad_ meet, we should have to go a long way to the left of the figure, which by this method is not necessary.

LVII

OF MEASURING POINTS

We now have to find certain points by which to measure those vanishing or retreating lines which are no longer at right angles to the picture plane, as in parallel perspective, and have to be measured in a different way, and here geometry comes to our assistance.

Note that the perspective square _P_ equals the geometrical square _K_, so that side _AB_ of the one equals side _ab_ of the other. With centre _A_ and radius _AB_ describe arc _Bm'_ till it cuts the base line at _m'_. Now _AB_ = _Am'_, and if we join _bm'_ then triangle _BAm'_ is an isosceles triangle. So likewise if we join _m'b_ in the perspective figure will m'Ab be the same isosceles triangle in perspective. Continue line _m'b_ till it cuts the horizon in _m_, which point will be the measuring point for the vanishing line _AbV_. For if in an isosceles triangle we draw lines across it, parallel to its base from one side to the other, we divide both sides in exactly the same quantities and proportions, so that if we measure on the base line of the picture the spaces we require, such as 1, 2, 3, on the length _Am'_, and then from these divisions draw lines to the measuring point, these lines will intersect the vanishing line _AbV_ in the lengths and proportions required. To find a measuring point for the lines that go to the other vanishing point, we proceed in the same way. Of course great accuracy is necessary.

Note that the dotted lines 1,1, 2,2, &c., are parallel in the perspective, as in the geometrical figure. In the former the lines are drawn to the same point _m_ on the horizon.

LVIII

HOW TO DIVIDE ANY GIVEN STRAIGHT LINE INTO EQUAL OR PROPORTIONATE PARTS

Let _AB_ (Fig. 117) be the given straight line that we wish to divide into five equal parts. Draw _AC_ at any convenient angle, and measure off five equal parts with the compasses thereon, as 1, 2, 3, 4, 5. From 5C draw line to 5B. Now from each division on _AC_ draw lines 4,4, 3,3, &c., parallel to 5,5. Then _AB_ will be divided into the required number of equal parts.

LIX

HOW TO DIVIDE A DIAGONAL VANISHING LINE INTO ANY NUMBER OF EQUAL OR PROPORTIONAL PARTS

In a previous figure (Fig. 116) we have shown how to find a measuring point when the exact measure of a vanishing line is required, but if it suffices merely to divide a line into a given number of equal parts, then the following simple method can be adopted.

We wish to divide _ab_ into five equal parts. From _a_, measure off on the ground line the five equal spaces required. From 5, the point to which these measures extend (as they are taken at random), draw a line through _b_ till it cuts the horizon at _O_. Then proceed to draw lines from each division on the base to point _O_, and they will intersect and divide _ab_ into the required number of equal parts.

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

LX

FURTHER USE OF THE MEASURING POINT O

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

Let _ABCD_ (Fig. 120) be the given square; produce its sides _AB_, _DC_ till they meet at point _V_. From _D_ measure off on base any number of equal spaces of any convenient length, as 1, 2, 3, &c.; from 1, through corner of square _C_, draw a line to meet the horizon at _O_, and from _O_ draw lines to the several divisions on base line. These lines will divide the vanishing line _DV_ into the required number of parts equal to _DC_, the side of the square. Produce the diagonal of the square _DB_ till it cuts the horizon at _G_. From the divisions on line _DV_ draw diagonals to point _G_: their intersections with the other vanishing line _AV_ will determine the direction of the cross-lines which form the bases of other squares without the necessity of drawing them to the other vanishing point, which in this case is some distance to the left of the picture. If we produce these cross-lines to the horizon we shall find that they all meet at the other vanishing point, to which of course it is easy to draw them when that point is accessible, 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 points and additional squares. Note the working of the diagonals drawn to point _G_, in both figures.

LXI

FURTHER USE OF THE MEASURING POINT O

Suppose we wish to divide the side of a building, as in Fig. 123, or to draw a balcony, a series of windows, or columns, or what not, or, in other words, any line above the horizon, as _AB_. Then from _A_ we draw _AC_ parallel to the horizon, and mark thereon the required divisions 5, 10, 15, &c.: in this case twenty-five (Fig. 122). From _C_ draw a line through _B_ till it cuts the horizon at _O_. Then proceed to draw the other lines from each division to _O_, and thus divide the vanishing line _AB_ as required.

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

LXII

ANOTHER METHOD OF ANGULAR PERSPECTIVE, BEING THAT ADOPTED IN OUR ART SCHOOLS

In the previous method we have drawn our squares by means of a geometrical plan, putting each point into perspective as required, and then by means of the perspective drawing thus obtained, finding our vanishing and measuring points. In this method we proceed in exactly the opposite way, setting out our points first, and drawing the square (or other figure) afterwards.

Having drawn the horizontal and base lines, and fixed upon the position of the point of sight, we next mark the position of the spectator by dropping a perpendicular, _S ST_, from that point of sight, making it the same length as the distance we suppose the spectator to be from the picture, and thus we make _ST_ the station-point.

To understand this figure we must first look upon it as a ground-plan or bird's-eye view, the line V2V1 or horizon line representing the picture seen edgeways, because of course the station-point cannot be in the picture itself, but a certain distance in front of it. The angle at _ST_, that is the angle which decides the positions of the two vanishing points V1, V2, is always a right angle, and the two remaining angles on that side of the line, called the directing line, are together equal to a right angle or 90 deg. So that in fixing upon the angle at which the square or other figure is to be placed, we say 'let it be 60 deg and 30 deg, or 70 deg and 20 deg', &c. Having decided upon the station-point and the angle at which the square is to be placed, draw TV1 and TV2, till they cut the horizon at V1 and V2. These are the two vanishing points to which the sides of the figure are respectively drawn. But we still want the measuring points for these two vanishing lines. We therefore take first, V1 as centre and V1T as radius, and describe arc of circle till it cuts the horizon in M1, which is the measuring point for all lines drawn to V1. Then with radius V2T describe arc from centre V2 till it cuts the horizon in M2, which is the measuring point for all vanishing lines drawn to V2. We have now set out our points. Let us proceed to draw the square _Abcd_. From _A_, the nearest angle (in this instance touching the base line), measure on each side of it the equal lengths _AB_ and _AE_, which represent the width or side of the square. Draw EM2 and BM1 from the two measuring points, which give us, by their intersections with the vanishing lines AV1 and AV2, the perspective lengths of the sides of the square _Abcd_. Join _b_ and V1 and dV2, which intersect each other at _C_, then _Adcb_ 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 a small angle, such as 10 deg on one side, and 80 deg on the other, then the size of the diagram becomes so large that it has to be carried out on the floor of the studio with long strings, &c., which is a very clumsy and unscientific way of setting to work. The architects in such cases make use of the centrolinead, a clever mechanical contrivance for getting over the difficulty of the far-off vanishing point, but by the method I have shown you, and shall further illustrate, you will find that you can dispense with all this trouble, and do all your perspective either inside the picture or on a very small margin outside it.

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

LXIII

TWO METHODS OF ANGULAR PERSPECTIVE IN ONE FIGURE

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

In each case we make the square _K_ the same size and at the same angle, its near corner being at _A_. It must be seen that by whichever method we work out this perspective, the result is the same, so that both are correct: the great advantage of the first or geometrical system being, that we can place the square at any angle, as it is drawn without reference to vanishing points.

We will, however, work out a few figures by the second method.

LXIV

TO DRAW A CUBE, THE POINTS BEING GIVEN

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

Let _A_ be the nearest angle touching the base. Draw AV1, AV2. From _A_, raise vertical _Ae_, the height of the cube. From _e_ draw eV1, eV2, from the other angles raise verticals _bf_, _dh_, _cg_, to meet eV1, eV2, fV2, &c., and the cube is complete.

LXV

AMPLIFICATION OF THE CUBE APPLIED TO DRAWING A COTTAGE

Note that we have started this figure with the cube _Adhefb_. We have taken three times _AB_, its width, for the front of our house, and twice _AB_ for the side, and have made it two cubes high, not counting the roof. Note also the use of the measuring-points in connexion with the measurements on the base line, and the upper measuring line _TPK_.

LXVI

HOW TO DRAW AN INTERIOR AT AN ANGLE

Here we make use of the same points as in a previous figure, with the addition of the point _G_, which is the vanishing point of the diagonals of the squares on the floor.

From _A_ draw square _Abcd_, and produce its sides in all directions; again from _A_, through the opposite angle of the square _C_, draw a diagonal till it cuts the horizon at _G_. From _G_ draw diagonals through _b_ and _d_, cutting the base at _o_, _o_, make spaces _o_, _o_, equal to _Ao_ all along the base, and from them draw diagonals to _G_; through the points where these diagonals intersect the vanishing lines drawn in the direction of _Ab_, _dc_ and _Ad_, _bc_, draw lines to the other vanishing point V1, thus completing the squares, and so cover the floor with them; they will then serve to measure width of door, windows, &c. Of course horizontal lines on wall 1 are drawn to V1, and those on wall 2 to V2.

In order to see this drawing properly, the eye should be placed about 3 inches from it, and opposite the point of sight; it will then stand out like a stereoscopic picture, and appear as actual space, but otherwise the perspective seems deformed, and the angles exaggerated. To make this drawing look right from a reasonable distance, the point of distance should be at least twice as far off as it is here, and this would mean altering all the other points and sending them a long way out of the picture; this is why artists use those long strings referred to above. I would however, advise them to make their perspective drawing on a small scale, and then square it up to the size of the canvas.

LXVII

HOW TO CORRECT DISTORTED PERSPECTIVE BY DOUBLING THE LINE OF DISTANCE

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

To redraw the whole figure double the size, including the station-point, would require a very large diagram, that we could not get into this book without a folding plate, but it comes to the same thing if we double the distances between the various points. Thus, if from _S_ to _G_ in the small diagram is 1 inch, in the larger one make it 2 inches. If from _S_ to M2 is 2 inches, in the larger make it 4, and so on.

Or this form may be used: make _AB_ twice the length of _AC_ (Fig. 130), or in any other proportion required. On _AC_ mark the points as in the drawing you wish to enlarge. Make _AB_ the length that you wish to enlarge to, draw _CB_, and then from each division on _AC_ draw lines parallel to _CB_, and _AB_ will be divided in the same proportions, as I have already shown (Fig. 117).

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

LXVIII

HOW TO DRAW A CUBE ON A GIVEN SQUARE, USING ONLY ONE VANISHING POINT

_ABCD_ is the given square (Fig. 131). At _A_ raise vertical _Aa_ equal to side of square _AB'_, from _a_ draw _ab_ to the vanishing point. Raise _Bb_. Produce _VD_ to _E_ to touch the base line. From _E_ raise vertical _EF_, making it equal to _Aa_. From _F_ draw _FV_. Raise _Dd_ and _Cc_, their heights being determined by the line _FV_. Join _da_ and the cube is complete. It will be seen that the verticals raised at each corner of the square are equal perspectively, as they are drawn between parallels which start from equal heights, namely, from _EF_ and _Aa_ to the same point _V_, the vanishing point. Any other line, such as _OO'_, can be directed to the inaccessible vanishing point in the same way as _ad_, &c.

_Note._ This is only one of many original figures and problems in this book which have been called up by the wish to facilitate the work of the artist, and as it were by necessity.

LXIX

A COURTYARD OR CLOISTER DRAWN WITH ONE VANISHING POINT

In this figure I have first drawn the pavement by means of the diagonals _GA_, _Go_, _Go_, &c., and the vanishing point _V_, the square at _A_ being given. From _A_ draw diagonal through opposite corner till it cuts the horizon at _G_. From this same point _G_ draw lines through the other corners of the square till they cut the ground line at _o_, _o_. Take this measurement _Ao_ and mark it along the base right and left of _A_, and the lines drawn from these points _o_ to point _G_ will give the diagonals of all the squares on the pavement. Produce sides of square _A_, and where these lines are intersected by the diagonals _Go_ draw lines from the vanishing point _V_ to base. These will give us the outlines of the squares lying between them and also guiding points that will enable us to draw as many more as we please. These again will give us our measurements for the widths of the arches, &c., or between the columns. Having fixed the height of wall or dado, we make use of _V_ point to draw the sides of the building, and by means of proportionate measurement complete the rest, as in Fig. 128.

LXX

HOW TO DRAW LINES WHICH SHALL MEET AT A DISTANT POINT, BY MEANS OF DIAGONALS

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

I must, however, point out the importance of the point _G_. In angular perspective it in a measure takes the place of the point of distance in parallel perspective, since it is the vanishing point of diagonals at 45 deg drawn between parallels such as _AV_, _DV_, drawn to a vanishing point _V_. The method of dividing line _AV_ into a number of parts equal to _AB_, the side of the square, is also shown in a previous figure (Fig. 120).

LXXI

HOW TO DIVIDE A SQUARE PLACED AT AN ANGLE INTO A GIVEN NUMBER OF SMALL SQUARES

_ABCD_ is the given square, and only one vanishing point is accessible. Let us divide it into sixteen small squares. Produce side _CD_ to base at _E_. Divide _EA_ into four equal parts. From each division draw lines to vanishing point _V_. Draw diagonals _BD_ and _AC_, and produce the latter till it cuts the horizon in _G_. Draw the three cross-lines through the intersections made by the diagonals and the lines drawn to _V_, and thus divide the square into sixteen.

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

LXXII

FURTHER EXAMPLE OF HOW TO DIVIDE A GIVEN OBLIQUE SQUARE INTO A GIVEN NUMBER OF EQUAL SQUARES, SAY TWENTY-FIVE

Having drawn the square _ABCD_, which is enclosed, as will be seen, in a dotted square in parallel perspective, I divide the line _EA_ into five equal parts instead of four (Fig. 135), and have made use of the device for that purpose by measuring off the required number on line _EF_, &c. Fig. 136 is introduced here simply to show 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 make it an oblong, as _ABEF_ (Fig. 136); for although we begin with a square we can extend it in any direction we please, as here shown.

LXXIII

OF PARALLELS AND DIAGONALS

To find the centre of a square or other rectangular figure we have but to draw its two diagonals, and their intersection will give us the centre of the figure (see 137 A). We do the same with perspective figures, as at B. In Fig. C is shown how a diagonal, drawn from one angle of a square _B_ through the centre _O_ of the opposite side of the square, will enable us to find a second square lying between the same parallels, then a third, a fourth, and so on. At figure _K_ lying on the ground, I have divided the farther side of the square _mn_ into 1/4, 1/3, 1/2. If I draw a diagonal from _G_ (at the base) through the half of this line I cut off on _FS_ the lengths or sides of two squares; if through the quarter I cut off the length of four squares on the vanishing line _FS_, and so on. In Fig. 137 D is shown how easily any number of objects at any equal distances apart, such as posts, trees, columns, &c., can be drawn by means of diagonals between parallels, guided by a central line _GS_.

LXXIV

THE SQUARE, THE OBLONG, AND THEIR DIAGONALS

Having found the centre of a square or oblong, such as Figs. 138 and 139, if we draw a third line through that centre at a given angle and then at each of its extremities draw perpendiculars _AB_, _DC_, we divide that square or oblong into three parts, the two outer portions being equal to each other, and the centre one either larger or smaller as desired; as, for instance, in the triumphal arch we make the centre portion larger than the two outer sides. When certain architectural details and spaces are to be put into perspective, a scale such as that in Fig. 123 will be found of great convenience; but if only a ready division of the principal proportions is required, then these diagonals will be found of the greatest use.

LXXV

SHOWING THE USE OF THE SQUARE AND DIAGONALS IN DRAWING DOORWAYS, WINDOWS, AND OTHER ARCHITECTURAL FEATURES

This example is from Serlio's _Architecture_ (1663), showing what excellent proportion can be obtained by the square and diagonals. The width of the door is one-third of the base of square, the height two-thirds. As a further illustration we have drawn the same figure in perspective.

LXXVI

HOW TO MEASURE DEPTHS BY DIAGONALS

If we take any length on the base of a square, say from _A_ to _g_, and from _g_ raise a perpendicular till it cuts the diagonal _AB_ in _O_, then from _O_ draw horizontal _Og'_, we form a square AgOg', and thus measure on one side of the square the distance or depth _Ag'_. So can we measure any other length, such as _fg_, in like manner.

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

To measure a length _Ag_ on the side of square _AC_, we draw a line from _g_ to the point of sight _S_, and where it crosses diagonal _AB_ at _O_ we draw horizontal _Og_, and thus find the required depth _Ag_ in the picture.

LXXVII

HOW TO MEASURE DISTANCES BY THE SQUARE AND DIAGONAL

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

In Fig. 144 the floor is divided into two large squares with their diagonals. Suppose we wish to draw a fireplace or a piece of furniture _K_, we measure its base _ef_ on _AB_, as far from _B_ as we wish it to be in the picture; draw _eo_ and _fo_ to point of sight, and proceed as in the previous figure by drawing parallels from _Oo_, &c.

Let it be observed that the great advantage of this method is, that we can use it to measure such distant objects as _XY_ 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 of perspective, 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 a given square in angular perspective, it is as well to have a ready and certain means of drawing that square without the elaborate setting out of a geometrical plan, as in the first method, or the more cumbersome and extended system of the second method. I shall therefore show you another method equally correct, but much simpler than either, which I have invented for our use, and which indeed forms one of the chief features of this book.

LXXVIII

HOW BY MEANS OF THE SQUARE AND DIAGONAL WE CAN DETERMINE THE POSITION OF POINTS IN SPACE

Apart from the aid that perspective affords the draughtsman, there is a further value in it, in that it teaches us almost a new science, which we might call the mystery of aspect, and how it is that the objects around us take so many different forms, or rather appearances, although they themselves remain the same. And also that it enables us, with, I think, great pleasure to ourselves, to fathom space, to work out difficult problems by simple reasoning, and to exercise those inventive and critical faculties which give strength and enjoyment to mental life.

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

Here, for instance, are two aspects of the same thing: the geometrical square _A_, which is facing us, and the perspective square _B_, which we suppose to lie flat on the table, or rather on the perspective plane. Line _A'C'_ is the perspective of line _AC_. On the geometrical square we can make what measurements we please with the compasses, but on the perspective square _B'_ the only line we can actually measure is the base line. In both figures this base line is the same length. Suppose we want to find the perspective of point _P_ (Fig. 146), we make use of the diagonal _CA_. From _P_ in the geometrical square draw _PO_ to meet the diagonal in _O_; through _O_ draw perpendicular _fe_; transfer length _fB_, so found, to the base of the perspective square; from _f_ draw _fS_ to point of sight; where it cuts the diagonal in _O_, draw horizontal _OP'_, which gives us the point required. In the same way we can find the perspective of any number of points on any side of the square.

LXXIX

PERSPECTIVE OF A POINT PLACED IN ANY POSITION WITHIN THE SQUARE

Let the point _P_ be the one we wish to put into perspective. We have but to repeat the process of the previous problem, making use of our measurements on the base, the diagonals, &c.

Indeed these figures are so plain and evident that further description of them is hardly necessary, so I will here give two drawings of triangles which explain themselves. To put a triangle into perspective we have but to find three points, such as _fEP_, Fig. 148 A, and then transfer these points to the perspective square 148 B, as there shown, and form the perspective triangle; but these figures explain themselves. Any other triangle or rectilineal figure can be worked out in the same way, which is not only the simplest method, but it carries its mathematical proof with it.

LXXX

PERSPECTIVE OF A SQUARE PLACED AT AN ANGLE NEW METHOD

As we have drawn a triangle in a square so can we draw an oblique square in a parallel square. In Figure 150 A we have drawn the oblique square _GEPn_. We find the points on the base _Am_, as in the previous figures, which enable us to construct the oblique perspective square _n'G'E'P'_ in the parallel perspective square Fig. 150 B. But it is not necessary to construct the geometrical figure, as I will show presently. It is here introduced to explain the method.

Fig. 150 B. To test the accuracy of the above, produce sides _G'E'_ and _n'P'_ of perspective square till they touch the horizon, where they will meet at _V_, their vanishing point, and again produce the other sides _n'G'_ and _P'E'_ till they meet on the horizon at the other vanishing point, which they must do if the figure is correctly drawn.

In any parallel square construct an oblique square from a given point--given the parallel square at Fig. 150 B, and given point _n'_ on base. Make _A'f'_ equal to _n'm'_, draw _f'S_ and _n'S_ to point of sight. Where these lines cut the diagonal _AC_ draw horizontals to _P'_ and _G'_, and so find the four points _G'E'P'n'_ through which to draw the square.

LXXXI

ON A GIVEN LINE PLACED AT AN ANGLE TO THE BASE DRAW A SQUARE IN ANGULAR PERSPECTIVE, THE POINT OF SIGHT, AND DISTANCE, BEING GIVEN.

Let _AB_ be the given line, _S_ the point of sight, and _D_ the distance (Fig. 151, 1). Through _A_ draw _SC_ from point of sight to base (Fig. 151, 2 and 3). From _C_ draw _CD_ to point of distance. Draw _Ao_ parallel to base till it cuts _CD_ at _o_, through _O_ draw _SP_, from _B_ mark off _BE_ equal to _CP_. From _E_ draw _ES_ intersecting _CD_ at _K_, from _K_ draw _KM_, thus completing the outer parallel square. Through _F_, where _PS_ intersects _MK_, draw _AV_ till it cuts the horizon in _V_, its vanishing point. From _V_ draw _VB_ cutting side _KE_ of outer square in _G_, and we have the four points _AFGB_, which are the four angles of the square required. Join _FG_, and the figure is complete.

Any other side of the square might be given, such as _AF_. First through _A_ and _F_ draw _SC_, _SP_, then draw _Ao_, then through _o_ draw _CD_. From _C_ draw base of parallel square _CE_, and at _M_ through _F_ draw _MK_ cutting diagonal at _K_, which gives top of square. Now through _K_ draw _SE_, giving _KE_ the remaining side thereof, produce _AF_ to _V_, from _V_ draw _VB_. Join _FG_, _GB_, and _BA_, and the square required is complete.

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

LXXXII

HOW TO DRAW SOLID FIGURES AT ANY ANGLE BY THE NEW METHOD

As we can draw planes by this method so can we draw solids, as shown in these figures. The heights of the corners of the triangles are obtained by means of the vanishing scales _AS_, _OS_, which have already 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 the scale _AS_, _OS_, we have made use of the vanishing lines _DV_, _BV_, to corroborate the scale, but they can be dispensed with in these simple objects, or we can use a scale on each side of the figure as _a'o'S_, should both vanishing points be inaccessible. Let it be noted that in the scale _AOS_, _AO_ is made equal to _BC_, the height of the box.

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 set down. Lines at any slope and at any angle can be drawn by this descriptive geometry.

The student can examine these figures for himself, and will understand their working from what has gone before. Here (Fig. 157) in the geometrical square we have a vertical plane _AabB_ standing on its base _AB_. We wish to place a projection of this figure at a certain distance and at a given angle 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 another perspective square lying flat, on which we have to stand our figure. By means of the diagonal of this flat square, horizontals from figure on side of cube, and lines drawn from point of sight (as already explained), we obtain the direction of base line _AB_, and also by means of lines _aa'_ and _bb'_ we obtain the two points in space _a'b'_. Join _Aa'_, _a'b'_ and _Bb'_, and we have the projection required, and which may be said to possess the third dimension.

In this other case (Fig. 158) we have a wedge-shaped figure standing on a triangle placed on the ground, as in the previous figure, its three corners being the same height. In the vertical geometrical square we have a ground-plan of the figure, from which we draw lines to diagonal and to base, and notify by numerals 1, 3, 2, 1, 3; these we transfer to base of the horizontal perspective square, and then construct shaded triangle 1, 2, 3, and raise to the height required as shown at 1', 2', 3'. Although we may not want to make use of these special figures, they show us how we could work out almost any form or object suspended in space.

LXXXIV

THE SQUARE AND DIAGONAL APPLIED TO CUBES AND SOLIDS DRAWN THEREIN

As we have made use of the square and diagonal to draw figures at various angles so can we make use of cubes either in parallel or angular perspective to draw other solid figures within them, as shown in these drawings, for this is simply an amplification of that method. Indeed we might invent many more such things. But subjects for perspective treatment will constantly present themselves to the artist or draughtsman in the course of his experience, and while I endeavour to show him how to grapple with any new difficulty or subject that may arise, it is impossible to set down all of them in this book.

LXXXV

TO DRAW AN OBLIQUE SQUARE IN ANOTHER OBLIQUE SQUARE WITHOUT USING VANISHING POINTS

It is not often that both vanishing points are inaccessible, still it is well to know how to proceed when this is the case. We first draw the square _ABCD_ inside the parallel square, as in previous figures. To draw the smaller square _K_ we simply draw a smaller parallel square _h h h h_, and within that, guided by the intersections of the diagonals therewith, we obtain the four points through which to draw square _K_. To raise a solid figure on these squares we can make use of the vanishing scales as shown on each side of the figure, thus obtaining the upper square 1 2 3 4, then by means of the diagonal 1 3 and 2 4 and verticals raised from each corner of square _K_ to meet them we obtain the smaller upper square corresponding to _K_.

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

LXXXVI

SHOWING HOW A PEDESTAL CAN BE DRAWN BY THE NEW METHOD

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

LXXXVII

SCALE ON EACH SIDE OF THE PICTURE

As one of my objects in writing this book is to facilitate the working of our perspective, partly for the comfort of the artist, and partly that he may have no excuse for neglecting it, I will here show you how you may, by a very simple means, secure the general correctness of your perspective when sketching or painting out of doors.

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 the rule. First of all note that line _HH_, drawn from one side of the picture to the other, is the horizontal line; below that is a wall and a pavement marked _aV_, also going from one side of the picture to the other, and being lower down at _a_ than at _V_ it runs up as it were to meet the horizon at some distant point. In order to form our scale I take first the length of _Ha_, and measure it above and below the horizon, along the side to our left as many times as required, in this case four or five. I now take the length _HV_ on the right side of the picture and measure it above and below the horizon, as in the other case; and then from these divisions obtain dotted lines crossing the picture from one side to the other which must all meet at some distant point on the horizon. These act as guiding lines, and are sufficient to give us the direction of any vanishing lines going to the same point. For those that go in the opposite direction we proceed in the same way, as from _b_ on the right to _V'_ on the left. They are here put in faintly, so as not to interfere with the drawing. In the sketch of Toledo (Fig. 164) the same thing is shown by double lines on each side to separate the two sets of lines, and to make the principle more evident.

LXXXVIII

THE CIRCLE

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

LXXXIX

THE CIRCLE IN PERSPECTIVE A TRUE ELLIPSE

Although the circle drawn through certain points must be a freehand drawing, which requires a little practice to make it true, it is sufficient for ordinary purposes and on a small scale, but to be mathematically true it must be an ellipse. We will first draw an ellipse (Fig. 167). Let _ee_ be its long, or transverse, diameter, and _db_ its short or conjugate diameter. Now take half of the long diameter _eE_, and from point _d_ with _cE_ for radius mark on _ee_ the two points _ff_, which are the foci of the ellipse. At each focus fix a pin, then make a loop of fine string that does not stretch and of such a length that when drawn out the double thread will reach from _f_ to _e_. Now place this double thread round the two pins at the foci _ff'_ and distend it with the pencil point until it forms triangle _fdf'_, then push the pencil along and right round the two foci, which being guided by the thread will draw the curve, which is a true ellipse, and will pass through the eight points indicated in our first figure. This will be a sufficient proof that the circle in perspective and the ellipse are identical curves. We must also remember that the ellipse is an oblique projection of a circle, or an oblique section of a cone. The difference between the two figures consists in their centres not being in the same place, that of the perspective circle being at _c_, higher up than _e_ the centre of the ellipse. The latter being a geometrical figure, its long diameter is exactly in the centre of the figure, whereas the centre _c_ and the diameter of the perspective are at the intersection of the diagonals of the perspective square in which it is inscribed.

XC

FURTHER ILLUSTRATION OF THE ELLIPSE

In order to show that the ellipse drawn by a loop as in the previous figure is also a circle in perspective we must reconstruct around it the square and its eight points by means of which it was drawn in the first instance. We start with nothing but the ellipse itself. We have to find the points of sight and distance, the base, &c. Let us start with base _AB_, a horizontal tangent to the curve extending beyond it on either side. From _A_ and _B_ draw two other tangents so that they shall touch the curve at points such as _TT'_ a little above the transverse diameter and on a level with each other. Produce these tangents till they meet at point _S_, which will be the point of sight. Through this point draw horizontal line _H_. Now draw tangent _CD_ parallel to _AB_. Draw diagonal _AD_ till it cuts the horizon at the point of distance, this will cut through diameter of circle at its centre, and so proceed to find the eight points through which the perspective circle passes, when it will be found that they all lie on the ellipse we have drawn with the loop, showing that the two curves are identical although their centres are distinct.

XCI

HOW TO DRAW A CIRCLE IN PERSPECTIVE WITHOUT A GEOMETRICAL _PLAN_

Divide base _AB_ into four equal parts. At _B_ drop perpendicular _Bn_, making _Bn_ equal to _Bm_, or one-fourth of base. Join _mn_ and transfer this measurement to each side of _d_ on base line; that is, make _df_ and _df'_ equal to _mn_. Draw _fS_ and _f'S_, and the intersections of these lines with the diagonals of square will give us the four points _o o o o_.

The reason of this is that _ff'_ is the measurement on the base _AB_ of another square _o o o o_ which is exactly half of the outer square. For if we inscribe a circle in a square and then inscribe a second square in that circle, this second square will be exactly half the area of the larger one; for its side will be equal to half the diagonal of the larger square, as can be seen by studying the following figures. In Fig. 170, for instance, the side of small square _K_ is half the diagonal of large square _o_.

In Fig. 171, _CB_ represents half of diagonal _EB_ of the outer square in which the circle is inscribed. By taking a fourth of the base _mB_ and drawing perpendicular _mh_ we cut _CB_ at _h_ in two equal parts, _Ch_, _hB_. It will be seen that _hB_ is equal to _mn_, one-quarter of the diagonal, so if we measure _mn_ on each side of _D_ we get _ff'_ equal to _CB_, or half the diagonal. By drawing _ff_, _f'f_ passing through the diagonals we get the four points _o o o o_ through which to draw the smaller square. Without referring to geometry we can see at a glance by Fig. 172, where we have simply turned the square _o o o o_ on its centre so that its angles touch the sides of the outer square, that it is exactly half of square _ABEF_, since each quarter of it, such as EoCo, is bisected by its diagonal _oo_.

XCII

HOW TO DRAW A CIRCLE IN ANGULAR PERSPECTIVE

Let _ABCD_ be the oblique square. Produce _VA_ till it cuts the base line at _G_.

Take _mD_, the fourth of the base. Find _mn_ as in Fig. 171, measure it on each side of _E_, and so obtain _Ef_ and _Ef'_, and proceed to draw _fV_, _EV_, _f'V_ and the diagonals, whose intersections with these lines will give us the eight points through which to draw the circle. In fact the process is the same as in parallel perspective, only instead of making our divisions on the actual base _AD_ of the square, we make them on _GD_, the base line.

To obtain the central line _hh_ passing through _O_, we can make use of diagonals of the half squares; that is, if the other vanishing point is inaccessible, as in this case.

XCIII

HOW TO DRAW A CIRCLE IN PERSPECTIVE MORE CORRECTLY, BY USING SIXTEEN GUIDING POINTS

First draw square _ABCD_. From _O_, the middle of the base, draw semicircle _AKB_, and divide it into eight equal parts. From each division raise perpendiculars to the base, such as _2 O_, _3 O_, _5 O_, &c., and from divisions _O_, _O_, _O_ draw lines to point of sight, and where these lines cut the diagonals _AC_, _DB_, draw horizontals parallel to base _AB_. Then through the points thus obtained draw the circle as shown in this figure, which also shows us how the circumference of a circle in perspective may be divided into any number of equal parts.

XCIV

HOW TO DIVIDE A PERSPECTIVE CIRCLE INTO ANY NUMBER OF EQUAL PARTS

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

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

The geometrical figure (177) will explain the construction of the perspective one by showing how the divisions are obtained on the line _AB_, which represents base of square, from the divisions on the semicircle _AKB_.

XCV

HOW TO DRAW CONCENTRIC CIRCLES

First draw a square with its diagonals (Fig. 178), and from its centre _O_ inscribe a circle; in this circle inscribe a square, and in this again inscribe a second circle, and so on. Through their intersections with the diagonals draw lines to base, and number them 1, 2, 3, 4, &c.; transfer these measurements to the base of the perspective square (Fig. 179), and proceed to construct the circles as before, drawing lines from each point on the base to the point of sight, and drawing the curves through the inter-sections of these lines with the diagonals.

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

Or we may adopt the method shown at Fig. 180, by taking quarter base of both outer and inner square, and finding the measurement _mn_ on each side of _C_, &c.

XCVI

THE ANGLE OF THE DIAMETER OF THE CIRCLE IN ANGULAR AND PARALLEL PERSPECTIVE

The circle, whether in angular or parallel perspective, is always an ellipse. In angular perspective the angle of the circle's diameter varies in accordance with the angle of the square in which it is placed, as in Fig. 181, _cc_ is the diameter of the circle and _ee_ the diameter of the ellipse. In parallel perspective the diameter of the circle always remains horizontal, although the long diameter of the ellipse varies in inclination according to the distance it is from the point of sight, as shown in Fig. 182, in which the third circle is much elongated and distorted, owing to its being outside the angle of vision.

XCVII

HOW TO CORRECT DISPROPORTION IN THE WIDTH OF COLUMNS

[Transcriber's Note:
The column referred to as "1" in the text is marked "S" in both
Figures.]

The disproportion in the width of columns in Fig. 183 arises from the point 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 3 is much wider than column 1.

In our second figure (184) is shown how this defect is remedied, by doubling the distance, or by counting the same distance as half, which is easily effected by drawing the diagonal from _O_ to 1/2-D, instead of from _A_, as in the other figure, _O_ being at half base. Here the squares lie much more level, and the columns are nearly the same width, showing the advantage of a long distance.

XCVIII

HOW TO DRAW A CIRCLE OVER A CIRCLE OR A CYLINDER

First construct square and circle _ABE_, then draw square _CDF_ with its diagonals. Then find the various points _O_, and from these raise perpendiculars to meet the diagonals of the upper square at points _P_, which, with the other points will be sufficient guides to draw the circle required. This can be applied to towers, columns, &c. The size of the circles can be varied so that the upper portion 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 the vanishing scale _POV_, which should be made the depth required, drop perpendiculars from the various points marked _O_, obtained by the diagonals, making them the right depth by referring them to the vanishing scale, as shown in this figure. This can be used for drawing garden fountains, basins, and various architectural objects.

C

APPLICATION OF PREVIOUS PROBLEM

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

CI

DORIC COLUMNS

It is as well for the art student to study the different orders of architecture, whether architect or not, as he frequently has to introduce them into his pictures, and at least must know their proportions, and how columns diminish from base to capital, as shown in this illustration.

CII

TO DRAW SEMICIRCLES STANDING UPON A CIRCLE AT ANY ANGLE

Given the circle _ACBH_, on diagonal _AB_ draw semicircle _AKB_, and on the same line _AB_ draw rectangle _AEFB_, its height being determined by radius _OK_ of semicircle. From centre _O_ draw _OF_ to corner of rectangle. Through _f'_, where that line intersects the semicircle, draw _mn_ parallel to _AB_. This will give intersection _O'_ on the vertical _OK_, through which all such horizontals as _m'n'_, level with _mn_, must pass. Now take any other diameter, such as _GH_, and thereon raise rectangle _GghH_, the same height as the other. The manner of doing this is to produce diameter _GH_ to the horizon till it finds its vanishing point at _V_. From _V_ through _K_ draw _hg_, and through _O'_ draw _n'm'_. From _O_ draw the two diagonals _og_ and _oh_, intersecting _m'n'_ at _O_, _O_, and thus we have the five points _GOKOH_ through which to draw the required semicircle.

CIII

A DOME STANDING ON A CYLINDER

This figure is a combination of the two preceding it. A cylinder is first raised on the circle, and on the top of that we draw semicircles from the different divisions on the circumference of the upper circle. This, however, only represents a small half-globular object. To draw the dome of a cathedral, or other building high above us, is another matter. From outside, where we can get to a distance, it is not difficult, but from within it will tax all our knowledge of perspective to give it effect.

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

CIV

SECTION OF A DOME OR NICHE

First draw outline of the niche _GFDBA_ (Fig. 193), then at its base draw square and circle _GOA_, _S_ being the point of sight, and divide the circumference of the circle into the required number of parts. Then draw semicircle _FOB_, and over that another semicircle _EOC_. The manner of drawing them is shown in Fig. 192. From the divisions on the circle _GOA_ raise verticals to semicircle _FOB_, which will divide it in the same way. Divide the smaller semicircle _EOC_ into the same number of parts as the others, which divisions will serve as guiding points in drawing the curves of the dome that are drawn towards _D_, but the shading must assist greatly in giving the effect of the recess.