The Theory and Practice of Perspective Part 16

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 c.u.mbersome 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 s.p.a.cE

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 s.p.a.ce, 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.

[Ill.u.s.tration: Fig. 145.]

[Ill.u.s.tration: Fig. 146.]

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 _AC_ is the perspective of line _AC_. On the geometrical square we can make what measurements we please with the compa.s.ses, 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.

[Ill.u.s.tration: Fig. 147.]

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.

[Ill.u.s.tration: Fig. 148 A.]

[Ill.u.s.tration: Fig. 148 B.]

[Ill.u.s.tration: Fig. 149 A.]

[Ill.u.s.tration: Fig. 149 B.]

Lx.x.x

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 _nGEP_ 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.

[Ill.u.s.tration: Fig. 150 A.]

[Ill.u.s.tration: Fig. 150 B.]

Fig. 150 B. To test the accuracy of the above, produce sides _GE_ and _nP_ of perspective square till they touch the horizon, where they will meet at _V_, their vanishing point, and again produce the other sides _nG_ and _PE_ 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 _Af_ equal to _nm_, draw _fS_ and _nS_ to point of sight. Where these lines cut the diagonal _AC_ draw horizontals to _P_ and _G_, and so find the four points _GEPn_ through which to draw the square.

Lx.x.xI

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.

[Ill.u.s.tration: Fig. 151.]

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.

Lx.x.xII

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.

[Ill.u.s.tration: Fig. 152.]

[Ill.u.s.tration: Fig. 153.]