[Ill.u.s.tration: Fig. 290.]

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

CLXV

ANGLES OF REFLECTION

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

[Ill.u.s.tration: Fig. 291.]

We have seen that the reflected line must be equal to the original line, therefore _mB_ must equal _Ma_. They are also at right angles to _MN_, the plane of reflection. We will now draw the visual ray pa.s.sing from _E_, the eye, to _B_, which is the reflection of _A_; and just underneath it pa.s.ses through _MN_ at _O_, which is the point where the visual ray strikes the reflecting surface. Draw _OA_. This line represents the ray reflected from it. We have now two triangles, _OAm_ and _OmB_, which are right-angled triangles and equal, therefore angle _a_ equals angle _b_. But angle _b_ equals angle _c_. Therefore angle _EcM_ equals angle _Aam_, and the angle at which the ray strikes the reflecting plane is equal to the angle at which it is reflected from it.

CLXVI

REFLECTIONS OF OBJECTS AT DIFFERENT DISTANCES

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

[Ill.u.s.tration: Fig. 292.]

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

[Ill.u.s.tration: Fig. 293.]

CLXVII

REFLECTION IN A LOOKING-GLa.s.s

I had noticed that some of the figures in t.i.tian's pictures were only half life-size, and yet they looked natural; and one day, thinking I would trace myself in an upright mirror, I stood at arm's length from it and with a brush and Chinese white, I made a rough outline of my face and figure, and when I measured it I found that my drawing was exactly half as long and half as wide as nature. I went closer to the gla.s.s, but the same outline fitted me. Then I retreated several paces, and still the same outline surrounded me. Although a little surprising at first, the reason is obvious. The image in the gla.s.s retreats or advances exactly in the same measure as the spectator.

[Ill.u.s.tration: Fig. 294.]

Suppose him to represent one end of a parallelogram _es_, and his image _ab_ to represent the other. The mirror _AB_ is a perpendicular half-way between them, the diagonal _eb_ is the visual ray pa.s.sing from the eye of the spectator to the foot of his image, and is the diagonal of a rectangle, therefore it cuts _AB_ in the centre _o_, and _AO_ represents _ab_ to the spectator. This is an experiment that any one may try for himself. Perhaps the above fact may have something to do with the remarks I made about t.i.tian at the beginning of this chapter.

[Ill.u.s.tration: Fig. 295.]

[Ill.u.s.tration: Fig. 296.]

CLXVIII

THE MIRROR AT AN ANGLE

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

[Ill.u.s.tration: Fig. 297.]

[Ill.u.s.tration: Fig. 298.]

Now suppose our mirror to be standing on a horizontal plane and on a pivot, so that it can be inclined either way. Whatever angle the mirror is to the plane the reflection of that plane in the mirror will be at the same angle on the other side of it, so that if the mirror _OA_ (Fig.

298) is at 45 to the plane _RR_ then the reflection of that plane in the mirror will be 45 on the other side of it, or at right angles, and the reflected plane will appear perpendicular, as shown in Fig. 299, where we have a front view of a mirror leaning forward at an angle of 45 and reflecting the square _aob_ with a cube standing upon it, only in the reflection the cube appears to be projecting from an upright plane or wall.

[Ill.u.s.tration: Fig. 299.]

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

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

[Ill.u.s.tration: Fig. 300.]

CLXIX

THE UPRIGHT MIRROR AT AN ANGLE OF 45 TO THE WALL

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

[Ill.u.s.tration: Fig. 301.]

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

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

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

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