in the virtual radiant r. Then the divergence of rPE and rpe is the same as the divergence of RPE and Rpe, and the distance being estimated by the divergence, the supposed distance behind the mirror is equal to the real distance in front of it.

As we must turn half round to see ourselves in the glass," our right and left sides are interchanged, and the whole picture, though not inverted, is reversed; and some curious effects follow. We see ourselves writing with the left hand, and in the same direction with Hebrew or Arabic. The printing of an open book will all run the wrong way. We hold the knife in the left hand, the fork in the right. We cannot shake hands properly with ourselves (I have known this statement cause great astonishment); and where, as is often the case, the two sides of the face are not perfectly alike, we never 66 see ourselves as others see us."

Many ingenious deceptions may be produced by large plane mirrors. They are often employed to increase the apparent size of shops and rooms. Two placed against opposite walls will reflect the back and front of intervening objects alternately to an interminable distance; and a small room may be turned into a long gallery; but the loss of light, which is considerable in each

reflection, soon makes the images dim. Setting the mirrors at an angle with each other produces curious varieties of effect, differing according to the angle. If two long narrow mirrors are placed face to face, and their lower edges kept close while their tops are gradually separated, a position will be found where an eye beyond one end will see them reflect each other into a complete circle, and any object beyond them at the other end will be reflected backwards and forwards till it appears in six places, and if it is of suitable form will produce a beautiful hexagonal, that is six-cornered pattern, which may be changed in a moment by moving the object, the images at the same time moving in opposite directions. This beautiful little instrument, the discovery of Dr. Brewster, is called the Kaleidoscope. It has been very serviceable to designers of patterns; and was at first so popular that 200,000 are said to have been sold in London and Paris during three months.

The goodness of a looking-glass depends chiefly on the evenness of its surface, any want of truth in which displaces the reflected rays irregularly, and distorts the images. Such a fault is readily detected by looking along it as obliquely as possible, when the increase of the angles renders an error in them more apparent.

We may now go on to our second case, Reflection from a Concave mirror. By this we mean part of a hollow sphere or globe, polished in the inside, such as a very regularly formed bowl or basin, or a silvered watch-glass, which, if not of the modern flat shape, makes a good concave mirror for its size. These, if we suppose them cut across in half so as to be viewed sideways, would show as parts of circles, and will appear as such in our diagrams. We shall be surprised perhaps at the difference which we shall find from having bent our plane mirror into a hollow one, even if ever so shallow, and the study will be very interesting. There will be no change-there never is in the law of equal angles of incidence and reflection; but the position of the Ps will be entirely altered. Every sphere, and every circle, has its centre, from which it is formed in practice, and which is at the same distance from every part of it. This we shall call and mark C. All straight lines from C to the outside, or circumference, which are called radii (singular, radius), are of course equal; and by what we have already seen, each of them may be considered P at the point where it touches the circle, or surface of the sphere. Now, to show how reflection will take place at a concave surface, we are not

going to use a candle. It would be convenient in some respects, and will do us good service by and by, but for the present we want something much smaller-a mere point if we could get it; a pin's head would do if we could keep it shining at a white heat. This is to be our R. Place it at C. What will become of its light? All of it will fall at right angles to the surface; it will go and come back in P, returning to C again. Now move R a little way from C towards the mirror. This must be of course along some radius; the most convenient will be that whose other end falls in the middle of the mirror. The light along this radius, being in P, will be reflected straight back; but what falls anywhere else will make a small angle RPC inside P, and be reflected at an equal angle CPF outside it. And since P is anywhere, all over the mirror, all the reflected rays will converge to the same point 1, cross there, and diverge again. This meetingpoint is called the Focus of reflected rays (foci if in plural number); we have marked it F on this diagram. It always falls on a straight line through C and R. If that line, as in the present

1 Not precisely the same point, if the mirror is part of a true sphere. The difference, which increases with the angles, is called the spherical error, or spherical aberration; it is not material to us now, but is of consequence in reflecting telescopes.

case, goes to the middle of the mirror, it is called its Principal Axis, any other line through CR falling elsewhere being an Oblique (or Secondary)

Axis.

Now let us move R gradually along the axis towards the mirror. In proportion as we do so, the incident rays will make larger angles with CP (P being supposed anywhere) and the reflected

ones will go further out, and owing to the slanting position as regards the axis, F will move much. faster out than R does in. But here we must look to our diagram. Radiant moving from C through R to r, focus moves more rapidly from C through F to f, and so on, till radiant reaches a point half way between C and the mirror, where the rays are reflected parallel among themselves, and to the axis, and go off to an unlimited dis

tance.

Mark this point PF, and call it the