REV. THOMAS W. WEBB, M,A, OPTICS :. WITHOUT MATHEMATICS. BY THE REV. THOMAS WILLIAM WEBB, M.A., F.R.A.S. PREBENDARY OF HEREFORD CATHEDRAL ; AUTHOR OF "CELESTIAL OBJECTS FOR COMMON TELESCOPES." LITERATURE AND EDUCATION APPOINTED BY THE SOCIETY FOR PROMOTING CHRISTIAN KNOWLEDGE. LONDON: SOCIETY FOR PROMOTING CHRISTIAN KNOWLEDGE, NORTHUMBERLAND AVENUE, CHARING CROSS, W.C. ; 43, QUEEN VICTORIA STREET, E.G. ; 26, ST. GEORGE'S PLACE, HYDE PARK CORNER, s.w. BRIGHTON: 135, NORTH STREET. NEW YORK; E. & J. B. YOUNG & CO. OPTICS WITHOUT MATHEMATICS. I PROPOSE to give you a little information in the science of Optics. Optics means the know- ledge of the nature and properties of Light. It is a very difficult subject in its full extent, and requires a deep knowledge of mathematics. But much that is not only very useful but very interesting is within the reach of any one who will take a little pains ; and I think I can pro- mise him that he will not be disappointed. There is never any disappointment in the reve- rent study of the great works of GOD. And what was His first recorded work? He said, Let there be Light ! And well might He thus open His wonderful and beautiful creation. We are so accustomed to the blessing of Light that we are scarcely able to value it as we ought. But let us try to think what our condition would be were light to be withdrawn from us were the sun to be darkened in the midst of heaven, and every artificial source of light to fail. It is B 2 4 OPTICS WITHOUT MATHEMATICS. a question whether life itself could in such a case be long maintained, so intimate is the con- nection of light with all the processes of animal and vegetable existence ; but at any rate it could only be a condition of comparative misery. Or, to take another illustration ; we pity the blind, and well we may : and it is our duty to do everything to alleviate so sad a privation. But what if we were all blind together, even from our birth ? Surely we do not speak amiss in saying that Light is one of the very best and chiefest gifts of Him who is pleased to describe Himself under. the name of Light. To light we owe by far the greater part of all that is valu- able and all that is lovely in this most beautiful world. But what is Light? We may say that it is that by which we see. But what is its nature? The true answer is probably far beyond the capacity of such a poor ignorant creature as man. We know nothing of the true nature of gravity, or electricity, or chemical action, any more than of light. All we can do is to seek for such an explanation as will account best for the properties of light, and to content our- selves with that till we are permitted to find a better. Such an explanation is given in a very complete and remarkable manner by what OPTICS WITHOUT MATHEMATICS. 5 is called the Undulatory Theory of Light. This supposes light to consist of undulations or waves *, so minute and so rapid as to pass our power of conception. Still, we had better try. We may get some notion in this way. We can easily divide an inch into 10 parts. The common carpenter's rule will not help us, be- cause he wants half-quarters, or eighths. But we shall find the inch divided into tenths on the scales in the cases of mathematical instru- ments. Now let us take a straight narrow strip of paper or card, and by means of the scale, mark off upon it the tenth of an inch. This we can easily cut in half crosswise with a fine pair of scissors ; and with sharp eyes and steady fingers we may manage to cut one of these halves across again into 5 parts ; each of these hair-breadth strips will then be about T ^th of an inch wide. Now, who could cut one of these into 10 again? If it were done, we should scarcely see them without a microscope. And 1 The undulatory theory, previously suggested by Hooke, the able contemporary of Newton, more clearly developed by Huy- gens, Fresnel, and others, and established by the researches of Dr. Young, has gradually supplanted that advocated by Newton. The supposition of waves or vibrations implies of course the further supposition of a medium in which they exist. This medium is called the ether or luminiferous ether ; but its existence being only assumed for the sake of the theory, it need not be further noticed. 6 OPTICS WITHOUT MATHEMATICS. yet it would take 50 of the waves of light to make up the breadth of one of these almost invisible slips of TTnnrth of an inch each. Such is the average size of these little undulations, some being rather larger and some less, as we shall see hereafter ; but generally speaking an inch would hold 50,000 of them. But we must take care to think of them not as anything like long strips, or waves of water, but as mere points, as broad as long, and undulating in every conceivable direction, provided it is crossways to the direction of the ray or pencil of light that enters the eye ; for it is considered certain that lengthways, or straight from the source of light, there is no undulation at all. So much for size. Now we have to learn something about their rapidity. How many of these little vibrations are supposed to take place in a minute ? It is useless to think of minutes ; we must try seconds. Only first we must understand what a second is. It is not, as some may suppose, the tick of a watch or an ordinary time-piece ; the beats of a great tall house-clock are the only real seconds, 60 to a minute. Now could you count 100 between each of those slow ticks? No, nor 10 without a good deal of hurry. But we have got to fancy not 10, nor 100, but billions of little ticks OPTICS WITHOUT MATHEMATICS. 7 between every great tick. And what is a billion? A million millions. A figure with only 12 ciphers after it ! We shall have to suppose 600 with 12 ciphers after it. And like the measure of size, this is an average number ; and if some kinds of light-vibrations are rather slower, others are more rapid still. Are we really expected to admit such mar- vellous statements ? Yes, we are ; for this plain reason, that no other supposition has been found to account for all the properties of light, and we must choose the least amongst difficulties. We cannot believe impossibilities. But there is nothing impossible in these figures. Let us recollect, once for all, in studying the works of the Almighty and Allwise, that nothing is in- credible merely because it is astonishing : "... Greater and less In Him are not . . ." These most minute, most rapid, undulations are set in motion by everything that we can see. Light proceeds, not only from shining bodies, such as the sun or a candle, but from everything that surrounds us, excepting of course in the dark: for all that we see is visible only by the quantity of light, be it more or less, that travels from the object to the eye. The whole universe must therefore be full of these OPTICS WITHOUT MATHEMATICS. wonderful vibrations, crossing each other in every conceivable direction, yet without obstruction or confusion. A conception astonishing as it is incomprehensible ! And now I shall proceed to explain to you at some length the commoner and more intelligible properties of Light ; with a shorter notice of some others of less frequent occurrence. We shall have to hear then about the Transmission, the Reflection, the Refraction, and the Disper- sion of Light ; with some mention of its Double Refraction and Polarisation ; including some account of the more remarkable optical instru- ments ; and ending with its Absorption at last. First, then, about the Transmission, or, as it is sometimes called, the Propagation of Light. By this is meant that when these minute un- dulations have been once set going, they spread themselves all round in every direction. When we throw a stone into still water, we produce an unlimited disturbance, spreading itself on all sides in a succession of gradually enfeebled waves. So it is with light ; but with this im- portant difference, that we only perceive the disturbance of the water in a horizontal direc- tion, but the waves of light are propagated in every conceivable way, up and down as well as all around : and this spreading, once set up, OPTICS WITHOUT MATHEMATICS. 9 extends itself everywhere, and lasts as long as its cause. If I strike a lucifer match in the middle of a dark room, the undulations that I cause will fill it everywhere ; if it were in the midst of St. Paul's Cathedral the same thing would happen, only so wide a spreading of the undulations would render them very feeble at the ends of the building ; in the open air there would be a distance beyond which they would be too faint to be seen, though a telescope would still render them visible, because it would catch more of them than could be taken in by the eye. This travelling away of light from its source is called its Transmission or Propagation. But what we have to observe is that it is not abso- lutely instantaneous. For instance, if we were to watch the firing of a cannon at night on a hill TO miles away, we should not see the light at the very moment of the discharge ; it would take time, though an inconceivably short time, to reach us. We all know how the same kind of thing happens with sound. If a gun is fired 100 yards off, we always see the light before we hear the noise ; and in a thunder- storm, unless we are quite close to the discharge which very seldom happens, but if it does we are not likely to forget it the flash is com- monly seen so long before the peal begins that 10 OPTICS WITHOUT MATHEMATICS. we may judge of our distance by the interval, 5 seconds showing rather more than a mile. Light and sound, in fact, both consist of un- dulations ; but those of light are much more minute, and travel faster than those of sound ; so much faster that if we are 10 miles from a storm we see the flash almost, though not quite, instantaneously, but the thunder will not begin till three quarters of a minute after. How- ever, though the velocity with which light travels is something astonishing, it is, as we have said, not absolutely imperceptible ; and though light does not take a sensible time to pass through 60 miles upon the earth, yet in those enormous distances which lie between the earth and the heavenly bodies its transmission from point to point becomes very perceptible. Astronomers have proved that the sun's light takes more than 8 minutes in crossing the 92,000,000 of miles which separate him from the earth ; so that if he were suddenly extinguished we should still see him for 8 minutes, till his latest ray had reached us ; and if he were as suddenly rekindled, we should not know it here for 8 minutes, till his fresh beams reached us again. And so inconceivably great are the distances of most of what we call the fixed stars, that their light must be TO, 20, 50, or 100 years in reaching us, so that we OPTICS WITHOUT MATHEMATICS. II cannot and do not see them as they are now ; for all we know, they may have long ceased to exist ; we only see them as they were so many years ago. This velocity of the transmission of light, or the speed with which it travels from point to point, is found to be about 186,060 miles in a second ! We must take care, however, not to confuse two entirely distinct things the rapidity of the undulations which form light, and the velocity of its transmission, when formed, from place to place. Something like the going of a watch when we journey by railway; the rate of ticking and the rate of the train in which the watch travels are perfectly distinct things, though both go on at the same time. The comparison fails however in one respect ; the watch goes when the train stands still, but the undulations of light and its transmission are inseparable. Nor in- deed is its transmission like the advance of a train, but more like the waving of a corn-field in a windy day. We have done with the transmission of light, because it is far too rapid to affect our ex- planation of its other properties. But before proceeding further, we must master some of the words that we shall have occasion to use. A ray of light. This means a line or stream of 12 OPTICS WITHOUT MATHEMATICS. undulations following one another at the extra- ordinary rate I have told you of, and always 1 going on in a perfectly straight direction. Sound, as we all know, turns corners ; light, in an ordinary sense, never does. The slenderer we conceive the rays to be, the better : we sometimes speak of a number of them side by side as a pencil of rays. The point from which the ray starts is called the radiant point, or for brevity the radiant ; and every object that we see emitting light from its whole surface, is con- ceived to be made up of, or to contain, an infinite number of radiants. Rays are either parallel, diverging, or converging. Parallel rays keep always the same distance apart, and never meet either way. Diverging rays are such as issue from the same radiant, constantly increas- ing their mutual distance ; but if the radiant is extremely far off, as sun, moon, and stars, or even the tops of distant hills, the rays in even a considerable pencil are so nearly parallel that they may be considered such. Converging rays meet in one point, or at least tend towards it. If they cross in that point and proceed further they become afterwards diverging. Medium (me~ dia if more than one) : this is anything that light 1 Not literally always, but the exceptions do not come under our notice at present. OPTICS WITHOUT MATHEMATICS. 13 passes through ; anything that carries it forward on its way. The vast spaces of the sky, that seem to us to be empty; air, water, ice, glass, crystals, jewels, all these are media; and these media, from allowing the passage of light, are called transparent. If we except the ' empty spaces of the sky (and there is a doubt even as to these), no media that we know of are perfectly transparent. Some are much more so than others ; but none transmit the whole of the light that they receive. Bodies which do not permit the passage of light are called opaque, but many even of these, when reduced to extreme thin- ness, allow a little to pass. What shall we find more opaque-looking than metals, such as gold and silver? One would as soon think of " see- ing through a millstone" as through a sovereign or a shilling ; yet when beaten out to extreme thinness even metals become partially trans- parent. You can try for yourself. If you breathe on a slip of glass, and lay it flat on a fragment of gold leaf so as to take it up, you will see the sun through it of a fine green colour ; or, if you take silver leaf, you will have a beautiful blue. And now we must make up our minds to learn a little about Angles. We need not alarm ourselves about them. We are pledged by our 14 OPTICS WITHOUT MATHEMATICS. title to avoid mathematics as much as possible ; but we cannot escape troubling ourselves with a very little of them. But indeed we have no right to call it " trouble ;" we shall find it useful in so many ways that we shall never regret master- ing the little that we shall find here. Angles are so often mentioned in common conversation, and sometimes so ignorantly, that I am anxious that you should know the real meaning of the word. An angle, then, is the mutual inclination or slope of two lines which meet one an- other, or would meet if they were carried far enough ; and which may be placed in any position whatever. And one angle is said to be greater or smaller than another, according as the slope is greater or smaller. The size of the angle has nothing to do with the length of the lines ; for the longest lines may form the smallest angle. Here are various specimens of angles, the smallest of which is the last but one, because the lines have the least mutual slope ; as the last is the largest for the opposite reason. OPTICS WITHOUT MATHEMATICS. 15 The dots show a case in which the lines, though cut short, may be said to make an angle. Angles are named by the letters that de- scribe the lines ; thus the angle formed at C by the lines AC, BC is called the angle ACB or BCA. If two straight lines cut through each other in any way, there will be of course 4 angles, and the opposite an- gles will be equal, as in this figure, where the angle A CB is equal to DCE, and A CD to BCE. But if two straight lines stand square to each other, as we say, so as to make the 4 angles all equal, then each of them is called a right angle, and each of the lines is said to be at right angles with (or to) the other, or to be perpendicular to it ; and if we rub out all the rest, so as to leave but two halves of the lines, they are still said to be at right angles or perpendicular to each other. And now we have prepared ourselves to understand one of the principal properties of Light, namely, Reflection. This means its being turned back by a polished surface. When a ray 1 6 OPTICS WITHOUT MATHEMATICS. of light meets such a surface, or in optical language is incident upon it, a portion of it, larger or less according to the nature and state of the surface, will be turned back. Even if the surface should be that of a transparent medium J , this reflection will take place. And it follows an invariable rule. Suppose a straight line drawn at right angles to the reflecting surface at the point where any ray falls. This we shall call the perpendicular, and for brevity express it by the letter P. Then the angle which the incident light makes with P will be equal to the angle which it makes again with P on the other side of it after reflection. Or, more briefly, the angle of incidence is equal to the angle of re- flection. And these angles are in the same plane. We must mind our spelling, and not write " plain." The words are akin, but plane has an especial meaning. It means a perfectly flat surface, great or small, and lying in any direction. A swing looking-glass, which may be set at any slope, and turned in all directions, is a complete illustration of what mathematicians mean by a plane ; only it must be capable of being extended, if necessary, to any conceivable distance. So when we say that the angles of incidence and reflection are in the same plane 1 There is an exception, but it would not be intelligible here. OPTICS WITHOUT MATHEMATICS. 17 with P, we mean that a perfectly flat card would pass through all three. But if the reflecting surface should .not be plane, but concave or hollow like a cup, or convex, that is, rounding, what will happen then? The same law will hold good. An ex- cessively short portion of a curved line may be considered straight for practical purposes, and therefore the radius, or straight line drawn from the centre of a circle to its outer edge or cir- cumference, or through it, is considered to be everywhere perpendicular to that circumference, inside or out ; and the angles of incidence and reflection will be equal on each side of it as before. Let me show you all this in three diagrams. In the first, CD is a section or edge- ways view of a plane mirror, which will be hori- zontal if of water, or oil, or mercury, but in any other position would be of glass or metal. Pp is a perpendicular drawn from any point in it you please, AP a ray falling on that point at any conceivable angle ; then it will be C i8 OPTICS WITHOUT MATHEMATICS. reflected to B, so as to make the angles APp, pPB equal. The other two diagrams show how the same thing happens with concave and convex mir- rors, such as may be made of glass or polished metal. These three are all the cases of regular re- flection that we shall mention now. But there is something to be said about each of them. First, about the plane mirror. If parallel rays, such as sun- or moonlight, fall upon it, they will be reflected parallel as before. This will be evident from the consideration that in the first figure P represents any point in the mirror, and therefore AP and PB are any rays in the whole pencil. And since everything is seen in the last direc- tion of the ray, an eye placed in the reflected pencil will suppose that the rays come from behind the mirror. But next, if the rays falling on the plane mirror are divergent, their diver- gency will not be altered. For the whole pencil diverging from one radiant, all the rays in it OPTICS WITHOUT MATHEMATICS. IQ make angles among themselves, which cannot be changed by reflection, since it only doubles each of them alike, and they will all go on as before, but as though they diverged from a virtual or imaginary radiant, as far behind the mirror as the actual radiant is in front of it; and an eye receiving the reflected rays will refer them to that virtual radiant. With convergent rays the same thing occurs ; but they are never met with in nature, and we need not consider their artificial production here. We have thus mastered, almost unawares, the very curious and interesting exhibition in the Looking-glass, which would seem to us a very surprising one, were we not so familiar with it. It is only a plane mirror , though too often a very imperfect one, from defects in material or workmanship. Every radiant in front of it is seen by reflected rays, which enter the eye as though they diverged in the same way from a radiant placed behind it. And since every ob- ject is but a collection of innumerable radiants, each object in front of the mirror will have a corresponding collection of apparent radiants behind it, which is called its image; and so any number of actual objects whose rays are re- flected by the mirror will have as many corre- sponding images apparently on the other side C 2 20 OPTICS WITHOUT MATHEMATICS. of it, filling up a perfect scene to an eye so placed as to receive it. The apparent distance of the image behind the looking-glass is owing to our unconsciously acquired habit of judging of all dis- tances by the use of both eyes at once. Among near objects, the picture in the right eye is not exactly the same with that in the left, and the mental process which combines the two images into one representation enables us in some wonderful way to estimate the distance accu- rately. Had we but one eye, we should find that estimate 1 a very different affair, as any one may convince himself by trying to thread a needle with one eye shut. Now this difference between the two eye-pictures and their combination, being exactly the same in the reflected as in the unreflected rays, we assign the same distance to the image as to the object. A diagram may help us here. The right eye E sees the reflected ray PE from R (the radiant ; we shall keep to this let- ter) in the direction EP which it continues towards r\ the left eye e sees the ray^te, also from R, in the direction ep> which intersects EP OPTICS WITHOUT MATHEMATICS. 21 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 esti- mated 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 our- selves 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 " 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 alter- nately 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 22 OPTICS WITHOUT MATHEMATICS. reflection, soon makes the images dim. Setting the mirrors at an angle with each other pro- duces curious varieties of effect, differing accord- ing to the angle. If two long narrow mirrors are placed face to face, and their lower edges kept close while their tops are gradually sepa- rated, 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 back- wards 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 in- strument, 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. OPTICS WITHOUT MATHEMATICS. 23 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 sup- pose 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 24 OPTICS WITHOUT MATHEMATICS. 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 some- thing 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 to- wards 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 RFC 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 J , cross there, and diverge again. This meeting- point 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. OPTICS WITHOUT MATHEMATICS. 25 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 /, 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 9 and call it the 2,6 OPTICS WITHOUT MATHEMATICS. i Principal Focus. Here the image of the sun will always be formed. The distance from it to *, or to the surface (both equal), is called the Focal Length of the mirror ; it is of course half the radius. If R goes in still further than PF, the rays, still keeping equal angles with CP, diverge, as shown by the dotted lines, and come to no focus at all, though their direction, con- tinued backwards beyond the mirror, will meet the prolonged axis at v. Such an imaginary meeting-point is called a Virtual Focus, which will approach the mirror behind in proportion as R does in front. Now since the course of light is reversible, let us make R and F change places. R will be. behind the mirror, and vir- tual, when F is very near its front. This does not happen in nature, where rays never converge, but may be artificially produced. R at an un- limited distance, that is, emitting practically parallel rays, comes to PF\ and such are the rays of sun, moon, and stars. R drawing nearer, as from very distant hills through trees or houses at a moderate distance to others close at hand, will be matched by F moving from PF towards C. And since the relative positions of R and F on either side of C are invariable, depending on the equality of angles, the positions corre- sponding to each other are called Conjugate, that is, connected, Foci. There is not much more to tell about radiants and their foci. But now, instead of a radiant let us get an object, that is, anything containing a multitude of radiants, and we shall find something quite fresh and very interesting, though all following naturally from what we have just learned. In place of our imaginary shining pin's head, we will have a lighted taper, and darken the room sufficiently: then every point in our taper and its flame being a radiant will have its conjugate focus, and the assemblage of all these foci will produce a bright image, corresponding with the object. These pleasing experiments with images require however larger mirrors than may come in our way, and we can show the principle on paper. We have set our lighted taper in front of the mirror. The wick is in the principal axis, and 28 OPTICS WITHOUT MATHEMATICS. somewhere in that axis its conjugate focus will fall, and its image be formed. A straight line from A, the point of the flame, through C to the mirror, will be an oblique axis, and the ray being reflected straight back, will have a con- jugate focus somewhere in that axis. But where? That we can easily show. Take a point any- where else in the mirror, draw CP to it, make the angles APC CPa equal, then Pa will be the direction of the reflected light, meeting the oblique axis through AC in a\ and since P is any point whatever in the mirror, all the light from A will be reflected to its conjugate focus in a. Now let us see about the rays from the other end of the taper. We had better draw another figure to avoid confusion. Then we have the conjugate focus of Z at z t as the con- jugate focus of A has just been shown to be at a. OPTICS WITHOUT MATHEMATICS. 29 But the whole object from A to Z is made up of innumerable points, each of which is a separate R, and every one of these Rs has its conjugate focus between Z and a ; the whole object AZ is therefore exactly represented by a corresponding set of foci between a and 2, or optically speaking, as is the image of AZ. But it is upside down ! This may seem strange, but you will find that it necessarily follows from the crossing of all the axes at C. And it is no virtual meeting of direction only; it is a real crossing of a multitude of rays, though being in the air it requires something to make it visible. This can be done in several ways ; by making a smoke to rise through the space ; by causing it to fall in a glass of water tinged with milk ; or more readily by receiving it on a piece of card or paper ; or if the eye is placed in the course of the rays after their crossing, at a sufficient dis- tance, it v/ill at once be seen. It is a curious phenomenon. But there are other things to be noticed about it. Light being reversible and the foci conjugate, the object and the image might change places ; at least if the object, unlike a taper, would bear inversion; and they would differ in size, the furthest from the mirror being largest, as we can make evident thus OPTICS WITHOUT MATHEMATICS. Let AZ be any object ; then from the equal angles at P it follows that, looked at from P t the divergence of the rays AZ which limit the object, is equal to the divergence of as which limit the image ; but when two straight lines diverge, it is obvious that the longer they are, the further they are apart ; and that, just in proportion to their distance ; so that as will be larger than AZ m pro- portion as it is further from P. And the reverse if object and image change places. Also they will not be equal in brightness ; the furthest from P will be the faintest : and that not in the propor- tion of the distance, but much more, of the square of the distance, that is, the distance multiplied by itself. For such must be the case with every in- fluence or power bounded by straight lines di- verging from a centre, as this figure shows OPTICS WITHOUT MATHEMATICS. 31 Here B is a surface of any form, illuminated by rays from A. At the distance C, twice as far, it is plain that the same light, diverging on every side, will cover 4 times as much surface, and at Z>, 3 times as far, it will be spread over 9 times as much space, and so on : but since the whole quantity of light is the same through- out, if it is spread over 4 or 9 times the surface, it must be 4 or 9 times weaker, that is, in pro- portion as the square of the distance. Hence images grow rapidly faint as they are formed further from the mirror, and the reverse. The virtual images at the back of the mirror are not inverted, because the rays have not crossed at C: they are also larger than the object, being formed further from the mirror. Large concave mirrors may be employed in light-houses, or in night-signaling. For a bril- liant light being placed in PF 9 the reflected rays will go out nearly parallel to any assignable distance, and may be directed upon any special point by moving the mirror. As the reverse of this application, large con- cave mirrors may be employed to concentrate the sun's rays at PF, and produce surprising results in burning, fusing, and volatilising almost everything on which the solar image falls. Much has not been attempted in this direction, though 32 OPTICS WITHOUT MATHEMATICS. the modern substitution of silvered glass for metal would render such experiments more feasible. A mirror of 6 feet (72 inches) in diameter and 10 feet in focal length would form an image of the Sun i inch in diameter 1 , in which the heat would exceed the ordinary solar heat in the proportion of i 2 to 72 2 , more than 5000 times, and as in putting the solar heat as low as 1 00 Fahrenheit, we should much more than allow for loss of light in reflection, we could reckon upon the almost inconceivable temperature of at least 5 OO ) OOO ' Few sub- stances could be found to withstand such heat ; almost every known material would be melted and dissipated. Such a mode of concentrating heat has of late been employed with great suc- cess by M. Mouchot, in the clear Algerine sky, for distillery purposes, and as a substitute for steam-power. Many curious illusions may be produced by large concave mirrors ; the object and mirror being concealed, and the image received on smoke, or on some suitable screen, or the eye being placed in the reflected rays. 1 As the sun's rays are practically parallel it might be at first supposed that the focal image would be a point : and so it would be if the sun were a point. But he is a broad disc, composed of innumerable points, each one of which forms its image in the focus, so as to produce a corresponding disc there. OPTICS WITHOUT MATHEMATICS. 33 The Convex Mirror is the reverse of the con- cave in every respect. The Ps instead of con- verging will diverge from the centre, which will be behind the mirror, and the reflected rays will diverge from virtual foci behind it, instead of con- verging to real ones in front of it. It has "there- fore no power of forming a real image, since the rays never cross, and the virtual images behind or within its surface are erect, and always smaller than their objects outside. Hence it can represent in a small compass the whole interior of a room ; and was formerly much used as a piece of ornamental furniture. To these three cases of Plane, Concave, and Convex Mirrors we may add some other forms of polished surfaces, such as cylinders and cones, which are seldom met with ; and more familiar combinations of curves, the distorting effects of which we recognise when we see our faces re- flected by a spoon. But the same law is pre- served, even in its most perverted applications. Reflection however is not confined to polished D 34 OPTICS WITHOUT MATHEMATICS. surfaces ; which are not the most common. Every object around us, except such as may be self-luminous, or black from reflecting no light, is seen no otherwise than by the light which it reflects, and which from the roughness of its surface is diffused in every direction, and meets the eye in every position. This is called Irrcgu- lar> or Diffused Reflection. Not only solid and liquid bodies, but even some gases, are visible in this way; and the brightness and blueness of the sky by day, and the morning and evening twilight, are due to the reflective power of the air. But for this, the noon-day sky would be black, as it must be on the Moon, which has so little atmosphere ; and every object not in sun- shine, or not aided by reflection from neigh- bouring objects, would be in absolute darkness. Such is the case in a telescopic view of the moon, where the shadow of every mountain is as midnight ; while the shadows on earth, viewed from a distance, would be softened and lightened by that atmospheric reflection, which contributes a very material though little suspected amount of convenience and comfort to the existence of man. We now proceed to another of the principal properties of light Refraction. We must leave Reflection for the present as completely OPTICS WITHOUT MATHEMATICS. 35 behind as if we had never learned it, and apply ourselves to a new rule and fresh phaenomena. Light may not only be turned back, as we have seen, from a polished surface without entering, as in Reflection, but may enter in, and be bent up or down, right or left,' or in fact in any direction, in the very act of en- trance ; and this bending is called Refraction. Some of it is always reflected 1 ; but a portion passes in, and we are to learn which way it will go. Now if a ray passes from one transparent medium to another, it is always refracted or bent, if the media are of different density. By density we mean the amount of matter in a certain space ; so that one medium is more or less dense than another according to the quantity of matter it contains. So glass is denser than water, water denser than air. Many other transparent media might be named, alco- hol, spirit of turpentine, oil, ice, crystals, the diamond. But we shall keep to glass, water, and air; they will answer every purpose, with less risk of confusion. This then we are distinctly to understand, that whenever the density of a medium changes, the path of light within it changes ; it is re- fracted, or bent, into a fresh line, as straight 1 See however note on p. 16. D 2 36 OPTICS WITHOUT MATHEMATICS. as the first, and in the same plane with it l , so long as the same density continues. But which way will it be bent? This is the first thing we have to find. And it will send us back to our old friend P, carried into both media. Our previous rule holds good, that any ray coinciding with P goes on unchanged. But the least angle with P causes refraction, by the following rule. If the ray passes from a rarer into a denser medium (rare expressing the reverse of dense) it is refracted towards P : if from denser to rarer, from P\ or in other words, the larger angle the ray makes with P is always on the rarer side. If then a ray should pass out of air into water, which way would it be refracted? Towards P. If from glass into water? From P. If from water into air, it will be bent from P. And why? Because water is the denser of the two. And here, as in reflection, it is immaterial which way the ray is supposed to go. The rule which goes on further to decide the amount of refraction, that is, the proportion of the angle of the refracted to that of the incident ray, is a curious one ; but it will take you rather further into mathematics than you may like. If so, you may skip it. But I advise you not. 1 There is an exception, which will be mentioned hereafter. OPTICS WITHOUT MATHEMATICS. 37 A little trouble in such matters is sure to pay itself. Here then it is. Let us try. You know something about angles, but not about the way of measuring their size : only you know it is not by the length of the lines. There are several ways. The most natural is, by the amount of opening of the angle, the sides being of equal length, as we can always make them. Thus if the angle BAG is equal to CAD, we should know that BAD was twice as great as either of them. This method is of course quite correct, and is generally used ; but another mode of com- paring the size of angles is employed in certain cases. It is 5 by their sines. We must mind our spelling as before ; not * signs,' either in look or sense; but sines. And what is a sine? Take any angle, as BAG, and from B anywhere in one line draw BD perpendicular to the other line AC\ then BD is the sine, that is, for the length AB ; for if we take any other point in A AB, as E, then EF is the sine for the length AE. We might have drawn the perpendiculars from the line A C to AB as well, if we chose, but it would complicate our figure, and come to the same thing. Now it is easy to see that the 38 OPTICS WITHOUT MATHEMATICS. sines, though they increase with the angles, so that the larger angle has always the longer sine, yet do not increase as fast as the angles. Let us take for instance the angle BAD, made up as before of two equal angles. Here the opening of BAD is twice that of CAD, but BF, the sine of BAD, is not twice as long as CE the sine of CAD ; and the larger the angle, the less in propor- tion is the increase of its .... sine as compared with the sine of the smaller angle. We have taken equal angles as the easiest case, but the rule holds good in all. Of course in order to compare the sines, we must make the lines forming the angles of equal length : and this is done at once by making A the centre of a circle, and drawing the sines from its intersection with the lines. But why are we to take all this trouble, and learn such a roundabout way of measuring and comparing angles, instead of the simple mode of estimating them by the amount of their opening along the circle we have drawn, or as mathema- ticians would say, by the arcs, or portions of the circle, which they include? Because we are dealing with angles in their reference to Light, OPTICS WITHOUT MATHEMATICS. 39 and it has pleased the Creator of Light in His wisdom so to order its nature, that the law of its refraction depends on the proportion not of the openings, or arcs, of the angles, but of their sines. And we must study it in this way if at all. This then is the law of Refraction, ' which those who have read carefully the previous paragraphs will be in a position to understand. When light passes from one medium to another of different density, provided it does not fall perpendicularly, in which case there is no re- fraction, it is so bent into a fresh course that the sine of the angle of incidence bears a constant or unvarying ratio^ or proportion, to the sine of the angle of refraction. That is, so long as the media continue the same, whatever angle an incident ray may make with P (the perpendicular at that spot to the surface which bounds the two media), the refracted ray will go on in such a fresh angle, that the sine of the first angle, from the smallest to the greatest possible, shall bear a fixed proportion to the sine of the second : that proportion being ascer- tainable only by experiment, and varying with differing media, but being unvaried as long as they are unchanged. Take any two media of differing density; send a ray through both, making any angle you please with P, which 40 OPTICS WITHOUT MATHEMATICS. may be put anywhere, and of course must be drawn through the surface into both media : then if the length of the sine of any particular angle in the one medium should happen to be f of the length of the sine of the corresponding angle in the other medium, the same proportion of f will be maintained between the sines of all other pairs of angles, from the smallest to the largest, that the ray can make with P in the two media, P being anywhere you please. This would actually be very nearly the proportion of the sines, if the ray were passing out of water into air. Or again, another pair of media being taken, if the length of the sine of incidence were to the length of the sine of refraction in any particular angle as 2 to 3, the same proportion would hold between the sines of any other con- ceivable pair of angles, and in any position of P. And this in fact is nearly the proportion when a ray passes from glass into air. Or as the direction of light is reversible, if it passed from air into glass, the proportion of sines would also be reversed, and become as 3 to 2. I do not know that I can put this more plainly, so if you do not as yet see it, and really wish to under- stand the principle, I must ask you to go over it again and again till you do. You will be sure to master it at last. OPTICS WITHOUT MATHEMATICS. 41 But now something more comes out of this. You see that the proportion of sines in any two media being invariable might always be ex- pressed by a fraction, -f or f , or any other value. But there would be two great disadvantages in this way of expressing the ratios. One is, that in very few instances would such simple fractions be accurate enough, and we should have to em- ploy numerators and denominators of many figures each to express the true value. The other is, that we should for the most part find great difficulty in comparing the ratios expressed by such fractions without the troublesome process of reducing them to a common denominator. We can get rid of all this trouble, simplify the whole affair, and express all our results in a tabular form, merely by converting each common into the corresponding decimal fraction. These can be carried to any required minuteness, and arranged with perfect ease for comparison. Thus for the proportion which is only approximately expressed by f air into water we may write I *33^ 5 a very accurate value; or for f air into glass 1-525 1 . And so on through all the pairs of known transparent media. In order to obtain uniformity of standard, all these fractions are referred to a supposed vacuum as unity, whose 1 Varying according to the kind of glass. 42 OPTICS WITHOUT MATHEMATICS. density, in optical calculation, differs very little from that of air. If other proportions are wanted, as those of glass and water, or alcohol and quartz crystal, they are easily obtained by cal- culation. Such a decimal fraction, expressing the proportion of the sines of incidence and re- fraction into a denser medium from a vacuum, or practically from air, is called the Index of refrac- tion of that medium, or, with more conciseness than propriety, its Refractive Index ; and tables of such indices are to be found in works on the subject. Here follow a few of the most familiar Vacuum ... . Hydrogen ... . Oxygen ... . Atmospheric air Ice Water Alcohol ... . Fluor spar ... . Spirit of Turpentine Crown glass Quartz Ruby Sapphire oooooo 000138 000272 000294 39 336 372 434 475 525 548 779 794 Diamond 2-439 We have done with these abstruser matters at last. Now let us try to apply the knowledge we have obtained. OPTICS WITHOUT MATHEMATICS. 43 You know what happens when a ray passes (not along P) from a denser to a rarer medium. It will be bent away from P. Then if it reaches the eye, which sees everything in the direction of the entering light, the radiant will seem to be higher up than it is. A figure will show this ; where AB is the common surface of water and air, C any object you please be- low the water, E the eye, CDp, PDE the angles of incidence and refrac- tion made by the ray CDE with P prolonged to />, which we shall use to signify a second P, whether the continuation of the first or not. Then E will see C at F. You can show this by an easy experiment. Lay a sixpence in the middle of an empty basin ; step back from it till the coin is just hidden by the edge of the basin ; pour in water gently so as not to shift the coin, and you will find it come gradually into view. Or, the converse of this, set a candle in such a position with regard to a basin that the shadow of the edge shall fall half way across the bottom ; fill the basin gradually with water (best if made turbid with a few drops of milk or ink) and note how the shadow will recede from its place. The first 44 OPTICS WITHOUT MATHEMATICS. experiment shows at once how water, looked at in a sloping direction, appears only about J of its real depth ; which might be a source of danger ; and how a straight stick, or one's fingers, appear to be bent if they are put slanting into water. Now let us put another case. What will happen to a ray passing through a denser medium with a rarer (of the same degree of V rarity) on each side of it. Can you think of such a case? You need not look far for it. It is that of a common window. Let us con- sider a pane magnified and viewed in section or edgeways. Let AB\)Q the glass ; CP any ray falling upon it anywhere ; it will be refracted towards P, and pass on as Pp. Then, the same densities coming in reverse order, the angles will do the same, and the ray will go out of the glass as pc, parallel to CP continued by a dotted line to D. And so it OPTICS WITHOUT MATHEMATICS. 45 may be shown that all rays, whether parallel, di- verging, or converging, will pass through parallel- sided glass with no change of direction. There will however be a slight change of place, accord- ing to the thickness of the glass, because PD is shifted to pc. This is hardly noticed, excepting with very oblique vision, in a common window of thin glass ; but if a piece of thick plate glass is looked through obliquely, the displacement is at once evident. The irregular displacements and distortions seen in looking through common glass windows arise from other causes the want of flatness in the surfaces, or of uniform density in the material. In plate glass, which is prepared with much greater care, these for the most part disappear. This was the case of glass in air on both sides, with parallel surfaces which would have been the same, only with a different proportion of sines, if the window could be made of ice, or crystal, or any substance denser than air. Now we must see what would happen if the two faces of the glass, instead of being parallel, were in- clined to one another in an angle, to which form they can easily be ground and polished. They would of course come to an edge where they meet, opposite to which would be some kind of back ; it does not matter of what form. Such a OPTICS WITHOUT MATHEMATICS. block of glass or any transparent medium is called a Prism ; and these 4 would be section-views of dif- ferent kinds of prisms, the size of the angle or length of its sides being immaterial. Now what will happen to a ray pass- ing through a prism, supposed denser than the surrounding me- dium ? All will depend as usual on the directions of P and/, and these will not be parallel, as the surfaces are in- clined to one another. Ray A refracted towards P proceeds to p; refracted there from p goes on to C. The angle between AP and fC } the first and last directions of the ray, is the angle of devia- tion. It will be evident that this increases with the increase of either the angle of the sides or the density of the prism, or both. All this would of course be reversed in a prism formed of a rarer surrounded by a denser medium. You will readily see that any object viewed through a prism will appear considerably out of its place : and that in whatever direction the prism may be placed, edge up, down, or sideways, the final deviation will be the same, towards the thickest OPTICS WITHOUT MATHEMATICS. 47 part, but following the position of the prism, and exactly corresponding with it. This must be carefully attended to, as it will immediately lead us on to something else, as we proceed from plane to convex and concave refracting surfaces. Supposing that we were to place three prisms (or pieces of prisms the same in practice) of dense material, as glass, one above another. It is plain that the angles might be so propor- tioned as to bring the refracted or emergent rays all to one point ; and if such an arrange- ment were turned upside down, or in any side- way position, the effect would be the same, as of course would be shown by turning the page round. But we may as soon fancy 30 prisms as 3, or 30,000 if we please ; and then we should get so close an approach to a curved surface that we could not distinguish them in practice ; and a surface of glass, or crystal, or diamond, equally curved upwards, downwards, sideways, and in every possible direction, would be found to re- 48 OPTICS WITHOUT MATHEMATICS. fract all incident rays to one point. It has been ascertained, fortunately for the construction of optical instruments, most of which require large pencils of rays to be brought to a point, that a portion of a sphere, which is easily worked, answers very nearly, though not with mathe- matical correctness ; and pieces of glass so shaped that their surfaces form portions of a sphere are called Lenses. But if a lens with two convex and spherical surfaces would be the re- sult of an indefinite, or, as mathematicians speak, an infinite number of prisms arranged edge outwards as above, then if we were to make the prisms of the opposite form, the thickest having the most inclined sides and the outermost posi- tion, an infinite number of these would form a spherical lens as before, but of the reverse form, concave instead of convex, having opposite pro- perties, and making the rays, not to converge towards an actual point, but to diverge, as the dotted lines show, from a virtual or imaginary one. OPTICS WITHOUT MATHEMATICS. 49 There are therefore two distinct kinds of lenses, the convex and the concave, the convex refracting inwards, the concave outwards ; and four forms of each, shown edgeways thus: 8 where it will be noted, that for clearness of ex- planation the most curved side, or, as opticians term it, the deepest^ is of the same radius in every case ; though of course it need not have been so. No. I. Equi-convex, two slices, as it were, put together from spheres of the same size. No. 2. Unequally convex, or in opticians' language, " crossed," in which one side has a longer radius, that is, is a portion of a larger sphere, than the E 50 OPTICS WITHOUT MATHEMATICS. other 1 . Of this form of course there may be varieties innumerable, and we can suppose one of the sides gradually flattened, till at last when its radius becomes infinite, that is indefinitely great, it may be considered a plane. This will be No. 3, the plano-convex lens, which is the equi-convex slit in two. Now if we continue to push in, as it were, the flattened side, we shall make it more and more concave, as in No. 4, the Meniscus^ so called from a Greek word sig j nifying the Moon. Here we should find that so long as the convex surface has the shorter radius, its Ps, converging (or diverging as you choose to take it) the more rapidly of the two, will produce the greater angles and the stronger refraction, and the convex predominating over the concave action, the meniscus takes rank accordingly, with a convexity which is the difference, instead of the sum, of the two refrac- tions. In one curious case, when the curves are so proportioned that the rays fall perpen- dicularly upon the second surface, all the re- fraction is done by the first. If we were to deepen the concave till its radius were equal to that of the convex, we should get a watch- glass, a form with no optical property, except distortion towards the edges; like a piece of 1 Nos. i and 2 are both frequently called double convex. OPTICS WITHOUT MATHEMATICS. 51 plane parallel-sided glass bent into a curved shape. The 4 concave lenses ; 5- t^ e Equi-concave ; 6. the Unequally or Crossed concave^; 7. the Plano-concave ; and 8. the Convexo-concave (or concavo-convex), are so exactly the reverse of the corresponding convex lenses, that it will be a good exercise for you to trace out the character of each till you have got it all clear before you. We shall find much similarity between the action of the concave mirror and the convex lens, and between the convex mirror and the concave lens ; as the convex lens, like the con- cave mirror, is the more generally useful, we shall explain it more particularly. The point where the rays meet, and from which they after- wards diverge, is, as with a mirror, the focus, real or virtual as the case may be. The focus for parallel rays is PF t and its distance from the lens is the focal length, or as it is frequently ex- pressed, the focus of the lens. As the rays may pass through the lens either way, it has a PF on each side. Ps and axes, objects and images, are as in mirrors, only not on one but on both sides. There is however this great difference between reflection and refraction, which if we do not yet 1 Nos. 5 and 6 are both frequently called double concave. E 2 52 OPTICS WITHOUT MATHEMATICS. clearly see, we ought not to go further till we have mastered it. In reflection the angles of incidence and reflection are both in the same medium, and always equal. In refraction the angles of incidence and refraction are in media of different density, and are always unequal, according to a proportion between their sines which differs in different materials. Still, the cases are many, as we shall see, in which reflec- tion and refraction lead to similar results : the position, the inversion, the magnitude, and the brightness of the focal images follow the same rule in mirrors and lenses ; but with this great difference, that in reflection the position of P, coinciding with the radius of the sphere, and the equality of the angles, are sufficient to deter- mine every case ; but in refraction we have to take into account the proportion of the sines. You might work all this out for yourselves ; but I will try to make it plainer by an example, thus, taking an equi-convex lens, and tracing the course of a ray through it that is, of course, of any ray that can fall upon it. You will un- derstand that if it passes along the principal axis, coinciding with P, it suffers no refraction ; everywhere else it will ; at the first convex sur- face towards P, at the second from /, and the combined effect will bring it down, as in the OPTICS WITHOUT MATHEMATICS. 53 prism, with an angle of deviation, to a focus, the position of which, whether principal or con- jugate, on the axis, be it principal or secondary, will be determined not only by the amount of curvature (that is, the position of P) as in reflec- tion, but also by the refractive power of the medium. It so happens that in the case of an equi-convex lens of ordinary glass this will bring out PF at about the distance of the radius, a convenient but accidental coincidence ; with a denser medium, as diamond, the focal length with the same radii would be shorter; longer, with a rarer medium, as ice. And so as to all diverging rays, which as you know are emitted by all objects at ordinary distances ; these, making greater and greater angles with P as the radiant approaches the lens, will have their divergency less and less diminished, and conse- quently their focal intersection with the axis 54 OPTICS WITHOUT MATHEMATICS. more and more deferred, till the radiant reaches the PFof rays coming in the opposite direction when its rays on reaching the lens will pass on parallel. If the radiant is moved still nearer to the lens, the rays will pass the parallel state and diverge, but with a diminished angle 5 so as to point backwards to a virtual focus more distant than PF. And all this will, as you know, hold equally good reversed, if you reverse the direc- tion of the light. Now, in further illustration, let us suppose our equi-convex lens split in two edgeways. What will result ? Two plano-convex lenses, each half as deep, or strong, that is, having half as much refraction, and therefore about twice the original focal length, which will not be affected by their facing opposite ways. But should we get the same result from split- ting an. unequally convex lens? Certainly not : for though we have the same medium, and there- fore the same refractive index, or power, we have different radii for the two convex faces, that is different positions of P, and consequently different angles and foci, always increasing as the curvature diminishes, and therefore longer for the one half-lens than the other. If we could imagine glass soft enough to be compressed at pleasure, and if we could gradually flatten the OPTICS WITHOUT MATHEMATICS. 55 convex side of either half-lens without injuring its spherical form, we should gradually lengthen the focus, till at last, when it became plane, we should have only a piece of common plate-glass left. All this of course would be reversed with concave lenses. We shall now be able to solve some interest- ing problems with such lenses as may be much more easily procured than mirrors would be. Reading-glasses and eye-glasses, or old spec- tacle-glasses, such as may be found at an op- tician's or watchmaker's, will do very well. We may have a few failures at first, for want of suitable focal lengths, but perseverance is sure of its reward. i. To find PF, the focal length, of any lens. If convex, which will be known by its magnify- ing objects, hold it up so that the sun may shine through it, taking care that the rays may fall upon it as perpendicularly as possible. Now trace their progress by moving a card, held parallel to the lens, steadily away from its back. You will find the circle of light gradually di- minish till it ends in a very small brilliant spot, and then enlarges again. That small spot is the image of the sun in PF. No lens will show it perfectly clear from surrounding light, and the shorter the focus in proportion to the aperture. 56 OPTICS WITHOUT MATHEMATICS. or breadth of the lens, the worse in this respect ; but we shall not stop to account for this now. With a concave lens, which diminishes objects, we must adopt a roundabout process, because it has no other than a virtual focus l . If not already circular (as a near-sighted spectacle-glass) we make it practically so by sticking on it a piece of paper with a circular hole in it, of any con- venient size ; we must also fasten the lens into a piece of card or stiff paper, so as to stop off all surrounding light ; any bits of thread that may be passed over it for the purpose of security will not injure the experiment. Now draw on an- other card, very carefully, a circle exactly twice the breadth of that you have stuck on the glass, and move that card gradually away from the lens. You will find the circle of transmitted rays grow larger (and fainter also, the same quantity of light being spread over a larger surface) till it precisely fills the circle on the card ; then it is in PF, for the course of the diverging rays traced backwards would meet in PF as far in front of the lens. 2. To neutralise the refraction of any lens of either kind. Set as near it as possible another of the opposite kind, of equal focus. The rays 1 A focus may be formed through it by a more powerful convex lens ; but this cannot be called its own focus. OPTICS WITHOUT MATHEMATICS. 57 will then pass through as through a piece of parallel-sided glass. 3. To alter the focal length of any lens. If you wish it shortened, place close to it 1 another of the same kind ; if lengthened, another of the opposite kind, but less strong, that is, of longer focus than the first ; as the result will follow the strongest, that is, the one which with the same density has the deeper curves, or with the same curves has the greater density. 4. To exhibit the images in conjugate foci ; a very interesting as well as instructive experi- ment. Set a convex lens upright on some kind of support near the edge of a table, on a level with the flame of a candle which may be moved across the table towards or from the lens, there being no other light in the room. The rays passing through the lens are to be caught on a sheet of cardboard or stiff paper, moveable to dif- ferent distances. If the effect is confused, what- ever may be the distance of the card from the lens, the flame is too near the lens to produce an image, and must be moved further till we begin to catch the image on the card. Then there's our candle upside down! Then let me move the candle gently to or from the lens, and do 1 This condition is not necessary ; but if the lenses are sepa- rated the problem becomes more intricate. 58 OPTICS WITHOUT MATHEMATICS. you move the card so as to keep the image as sharp as possible. We shall find that within certain limits, as the one advances the other recedes, and the flame and its image will occupy the conjugate foci, and we shall be eye-witnesses of the facts that have been mentioned as to the varying size and brightness of the images. Un- fortunately we could not form them erect with- out turning the candle upside down. 5. To use a burning-glass. A convex lens being held in the sun's rays, an image of him will be formed in PF 9 corresponding with that of the concave mirror. If this falls on the hand, unless the lens is very small, the effect will soon be perceptible, and paper placed there will be charred and smoulder away, especially if the place has been rendered more absorbent of heat by a spot of ink. A large aperture and short focus are required for a very active burning glass ; the former to collect more heat, the latter to concentrate it in smaller compass ; for we must bear in mind that every image is larger in proportion to its distance from the lens ; the oblique axes drawn from the top and bottom of the object through the centre of the lens, and on which the image of those parts is formed, sepa- rating more and more in proportion to their dis- tance from the lens. Hence the solar image, OPTICS WITHOUT MATHEMATICS. 59 which is a mere point with a lens of 3 or 4 inches focus, becomes a disc of some size with one of as many feet, and unless the lens is larger in proportion, has not much burning power, the heat, like the light, being diffused over a larger area, increasing as the square of the distance. It would be possible to form a burning lens of ice ; a strange but easily intelligible modifica- tion of the experiment. Very large burning lenses have been made of glass ; the largest by Tschirn- hausen at Paris in 1687, 4 feet in diameter; another by Parker at London at the end of the last century, with a diameter of 2 feet 8J inches, and another of 13 inches between it and PF, to obtain a shorter focus and therefore hotter image, which was |-inch in diameter. The heat here was estimated equal to at least 2400 times that of the hottest sunshine. It burnt, melted, and destroyed everything. Its money value was 700 guineas ; its scientific value very con- siderable ; and it would hardly be imagined that it was presented to the Emperor of China l ! 6. To form a picture of an external scene. This is done, but in an imperfect way, and per- haps you have noticed it, by a small hole behind which there is a considerable dark space, and a screen of some kind at a distance to catch 1 A model of it may be seen in the South Kensington Museum. 60 OPTICS WITHOUT MATHEMATICS. the light coming through the hole. A key-hole in the outer door of a house with a dark passage behind it will show a curious though indistinct in- verted picture on a cross wall, of things outside the door ; it will be improved by setting a card with circular hole against the key-hole to get it into better form. This picture merely results from the crossing of small pencils of rays in the hole, the lower going to a higher place on the wall, and the reverse, any great amount of overlapping and confusion being prevented by the limited size of the hole the smaller the better, provided it admits light enough ; but the representation is never distinct, because the rays are not brought to points. But if we place at the key-hole a convex lens of long focus, of the kind called " preservers," the picture will be strikingly clear. Of course it will not be equally sharp every- where, for neither are all external objects, nor all parts of the wall or screen, equally distant from the lens on either side, and the conjugate foci are of course perfect only for one distance ; and if one part of the picture, to use a common expression, is " in focus," other parts will be " out of focus " : the general effect however is very pleasing; and the inversion of moving figures very amusing. This is the principle of the ap- paratus called the Camera Obscura, which is OPTICS WITHOUT MATHEMATICS. 6 1 sometimes made on a large scale for public ex- hibition, with a picture thrown on a table by means of a plane mirror outside on the roof so as to avoid turning the landscape upside down. The photographic Camera is also the same thing, the picture being formed by 'the lens inside the camera or box on a material capable of receiving and retaining the impression of light ; a process, the wonder of which has only been diminished by its familiarity. In the Camera Lucida^ which acts on the same prin- ciple, the picture is so arranged that pencil and paper can be seen with it so as to trace it, for the benefit of less competent draughtsmen. 7. To magnify near objects. You must have noticed long ago, in handling convex lenses, their property of magnifying : you shall now learn how to manage it to advantage. We shall say some- thing at a future time about the wonderful me- chanism of the eye ; at present it is enough to state that a good eye sees perfectly only by either parallel rays, or, in ordinary cases, by those di- verging from radiants more than 8 or 10 inches distant ; with either stronger divergence, or any kind of convergence, vision is indistinct. Now the apparent magnitude of every object, or the space it takes up in the eye, is in the inverse propor- tion of its distance, that is, becomes greater as 62 OPTICS WITHOUT MATHEMATICS. the distance is less ; and therefore if we want to see a thing minutely we look closer at it ; but if we get it too near, the rays will diverge too much to form a clear picture in the eye, and it be- comes confused. Can you think of any way of rectifying this ? A little consideration will show you that these diverging rays are in the same state as if they had passed through a deep con- cave lens, and you have learned, in Problem 3, how you might reduce the divergency by the use of a convex lens. Now you can bring your knowledge into action. Place a suitable lens in front of the eye, and you will reduce the diver- gency till the object is seen perfectly well, though very near the eye. But while you have thus, so to speak, sharpened it, you have magnified it at the same time. For suppose the nearest limit of your distinct vision was at 8 inches, then if you have brought it to 4 inches and yet kept it distinct, you see it twice as large ; if you have brought it to i inch with clearness, you will have it magnified 8 times, and so on. Or put it in another form. Your sight is good for parallel rays, for you can see the moon as a sharp circle, and the stars as the minutest points. But the wonderful structure of a flower or a feather is too minute for you ; you try to see it closer, but it only gets indistinct, because the OPTICS WITHOUT MATHEMATICS. 63 rays are too divergent. Put it then in PF of a convex lens ; those rays will be rendered parallel, and you will see every part sharply, and enlarged according to its nearness \ And thus we have come all at once on both the principle and the practice of the Simple Micro- scope > a convex lens, having in PF the object you wish to magnify, the power of which will be found by dividing the smallest distance at which you see the object perfectly, by the focal dis- tance of the lens. In practice we shall place the object on a stage, near PF, and make the distance from the lens variable by a screw ; for which there are two reasons ; one, that unless the object is very minute or thin, its different parts will not be all at the same distance from the lens, and will require fresh adjustment to bring them successively into PF \ the other, to suit different eyes, some of which may see better by 1 It may perhaps appear strange that divergent rays should be rendered parallel without changing the apparent size of the object due to its distance. But we must bear in mind that the pencils of rays diverging from every point in the object to the whole surface of the lens being all rendered parallel among them- selves, all take the direction of the central ray among them, and this, being in fact a secondary axis of the lens, passes through practically unrefracted : these central rays of every pencil cross- ing in the lens proceed to form the image in the eye, which will consequently, as far as size is concerned, be nearly the same as if no lens had been there. 64 OPTICS WITHOUT MATHEMATICS. diverging or converging than by parallel rays. A perfect eye will see well with any divergency between parallelism and 8 inches and some at a less distance and for all those angles our shifting stage makes provision, with of course some corresponding change of magnifying power. We have now got a beautiful instrument in our hands, a very simple apparatus, yet quite suffi- cient to give us a delightful insight into the wonders of Creation. But we shall soon find it is a very limited one. Such a contrivance has some serious drawbacks. Cannot you find them out ? You would soon experience them if you made the trial. You can magnify very nicely in this way to a small extent ; but this would rather sharpen than satisfy your appetite ; when you had seen what a power of 20 would do, you would long to go on to 200 ; but on this construction you would soon break down. The higher your power the deeper your curves and the smaller your lens, soon too small for convenience ; the darker too would be your image, the light being spread over so wide an area; and though this might in some measure be remedied by artificial illumination, yet the nearness of the object to the lens would make this troublesome ; and there would be other drawbacks, which you are not yet in a position to OPTICS WITHOUT MATHEMATICS. 65 understand, that would soon check your advance. Can you think of any remedy? You know enough already to manage it, but the way may not occur to you. You know that when rays cross in focus an image is formed, which, though invisible out of the course of the rays, to an eye which receives them is the exact counterpart of the object. And you know how a convex lens will form an image much larger than the object, by properly arranging conjugate foci ; in fact you have done this for yourself if you have followed out the experiments I gave you. Now what should hinder your making an object, so to speak, of this magnified image, and magnifying it over again ? Nothing is easier. Put the real object in front of a deep lens, which you may call the object-glass, so that the conjugate image shall be 10 times larger; then look at this image through another lens, which you may call the eye-lens, magnifying 10 times, and you get at once a power of loo 1 . And the object not being too close to the first lens, you can 1 The magnifying power of a microscope may be estimated in two ways ; either by the lengthening of lines, or by the enlarge- ment of area, which is of course the square of the linear measure. For if a line is made 10 times longer, the square of which it forms a side will of course contain 100 times more space. The mag- nifying power of a microscope is more frequently given in super- ficial measure ; but it is better for our purpose to reckon it in the linear form. F 66 OPTICS WITHOUT MATHEMATICS. readily illuminate it so as to make up in some measure for the loss of light. And so you have made a Compound Microscope \ of very primitive construction it is true, and subject to great drawbacks, but which, when improved in ways which you could not as yet understand, has done great service in revealing to us the incon- ceivable perfection of the minutest of the works of the Allwise Creator. The Magic Lantern is an adaptation of the same principles, in which the light of the lamp A, reflected as much as may be by the concave mirror Z?, is made by the lens C to converge on an inverted painting on the slide (seen endways) D ; this painting is placed in one of the conjugate foci of another lens , and received erect on a screen in the corresponding focus F. Combina- tions of lenses are used to improve the image, but the principle is represented here. Our last problem led to a very interesting result. The next will fairly equal or surpass it. 8. To magnify distant objects. Nothing is OPTICS WITHOUT MATHEMATICS. 67 more natural than the wish to see more of a remote prospect, or a ship at sea; and more especially to catch a glimpse of some of those wonderful things in the heavens, of which we read so much in books. I am going to show you how it is done. You cannot proceed, as in the last case, by trying to look through a convex lens. And you ought to be able to tell me why. It is because the object is too far off to be put in front of the lens in PF or any available conjugate focus. The rays from the heavenly bodies at their enormous distance are sensibly parallel ; and those from a far-off view, or in- deed from moderate distances, diverge too slightly to admit of being treated as in the simple microscope. All you could do is this. You may, as in the compound microscope, look at the image instead of the object. A spectacle lens of very long focus, one of those called No. i, or "preservers," would form an 'image in PF of any heavenly body large enough for the purpose : and if we bring our eye into the pencil of rays which have crossed at the focus, far enough back to see the image distinctly, say at 8 inches, we shall perceive a picture of the object magnified as much as the focal image exceeds the extremely minute picture which we had in the eye without using the lens. Magni- F 2, 68 OPTICS WITHOUT MATHEMATICS. fying power means the increased space that the focal image takes up in the interior of the eye, compared with that existing there under ordinary circumstances. The eye acts as a convex lens, but of so short a focus that it is not much considered in these questions : we may put it at about half an inch. Now we have learned that the magnitude of the focal image is in proportion to its distance from the lens : the image therefore formed by a lens of 8-inch focus will be 16 times larger than that in the eye. But the eye cannot see that image, or anything else, in contact with itself, being far within the limits of distinct vision ; it will therefore have to be withdrawn about 8 inches J , and then we can see the moon, for instance, clearly, but not magnified, the gain of size in the focal image formed by the lens being exactly counteracted by the loss resulting from removing the eye back to see it distinctly. But now let us take a lens of 1 6-inch focus; this will form us an image twice as large as the last ; but we can still keep our eye 8 inches from it, and therefore we shall have gained a magnifying power of 2, and if so we go on to focal lengths of 32 and 64 inches, we shall find our powers increase to 4 and 8 ; and thus we shall have ob- 1 Assumed as a general and convenient distance ; some eyes, though perfect, may see closer and produce a higher power. OPTICS WITHOUT MATHEMATICS. 69 tained a general rule, that for ordinary vision the magnifying power of a convex lens turned on celestial or distant terrestrial objects is equal to the focal length in inches divided by 8, or the least distance of distinct vision, whatever that may be. The image of course will be in- verted ; but this, though unpleasant for a ter- restrial view, is unimportant for the heavenly bodies. The arrangement would however be very inconvenient. We should require a lens of 20 feet focus to magnify only 30 times, and that would require a tube of the troublesome length of 20 feet 8 inches. Cannot our experience with the microscope help us here? Certainly. We can just as well magnify the focal image with a second lens as look at it with the eye, and then as we reduce the distance of 8 inches we shall increase the power, and a i-inch eye-lens would give on an object-lens of 20 feet a power of (30 x 8) 240, amply sufficient to give us an astonishing revelation of the glories of the firma- ment. That is, in theory. How disappointing it would be in practice we shall see hereafter. But at any rate we have thus been led on to the construction of an Astronomical Telescope ; the inverted image in that case being of no importance to the practised eye. In fact we might readily make such a telescope with two 70 OPTICS WITHOUT MATHEMATICS. convex lenses of 36 inches and i inch focus, set in a tin tube about 38 inches long, blackened in the inside, the eye-lens being placed in a sliding paper tube for adjustment, and set in i inch or more from the eye-end, to guide the eye to the rays and exclude disturbing light. We should have a power of 36 (that number being divided, and the quotient multiplied by 8), sufficient to show us the craters in the moon, and the satel- lites of Jupiter, and a little of the ring of Saturn; but we should soon be sensible of such defects as to make us long for a more perfect con- struction. For terrestrial purposes the inver- sion would have to be rectified ; and this would be done by two additional convex lenses, be- tween the object-lens and the eye-lens : we need not however particularly describe this con- trivance, which will be met with in taking any terrestrial telescope to pieces, and occasions its additional length. The astronomical telescope however was not invented in this its simplest form ; so simple that it is surprising that its dis- covery should have been so long delayed. As originally invented, or rather worked out, by Galileo, the eye-lens was concave, of shorter focus than the object-lens ; this being interposed in the converging rays before they reached PF 9 would render them parallel, and produce distinct OPTICS WITHOUT MATHEMATICS. ?I vision ; the power of magnifying being explained thus : Let A C, BD be the direction of rays proceeding from the top and bottom of any object ; these will be refracted by the eye-lens to E and F, so as to form an enlarged image in the interior of the eye. The concave and convex eye-lenses being the exact optical counterpart of each other, the magnifying power with the same focus will be identical 1 ; but the telescope will be shorter, by the difference instead of the sum of the focal lengths, and the image, as the rays do not cross, will not be inverted. Hence it is very suitable for terrestrial use, and is commonly employed for field or opera glasses in a binocular form. With high powers, however, the field of view is so very small that it is seldom resorted to for astronomical purposes. And now we have mastered the theory of the Refracting Telescope in its simplest forms. One important addition to this, as well as to the 1 The same mode of showing the magnifying power might have been adopted with the convex lens ; but the other was preferred, as explaining its amount more readily. 72 OPTICS WITHOUT MATHEMATICS. microscope, is the dividing the eye-lens, or ocular, into two, so as to obtain a much more perfect field of view ; a good deal more however must be done to improve it, as we shall hereafter see, before it can become the beautiful instrument of modern days. You have so frequently heard the Eye referred to as a kind of optical instrument that it is quite time that you should learn something about it ; but it can be but very little in comparison with the extent of this wonderful subject. It requires a volume to treat properly of one of the most astonishing evidences of Divine Wisdom. The eye is a ball, much larger than we may suppose from what we see of it. It is lodged in a well- defended cavity in the skull, so as to be in immediate communication with the brain ; and it is double, not only to provide for the possible loss of one, but to enable us to judge of the dis- tances of near objects from the fact that each eye sees its own picture from its own point of view, and the mind, combining them into one, is able to estimate their distance, as we shall pre- sently see. The eye- ball is composed of several coats ; one inside another : the outermost, or sclerotica^ a very hard tough membrane, the " white of the eye " : the front part of this becomes quite trans- OPTICS WITHOUT MATHEMATICS. 73 parent, and stands out as a portion of a smaller sphere ; this is called the cornea, A in the dia- gram, and is the most exposed part of the eye. Inside the sclerotica, except where it bulges out to form the cornea, is a perfectly black lin- ing, called the choroid coat : this darkens all the inside so as to ab- sorb all needless light, and causes, seen in the depths, the blackness of the hole we call the pupil, between C and c : this is a real open- ing into the inside, surrounded by a coloured circle called the iris, which joins on to the choroid, and is differently tinted in different persons. Im- mediately behind this is the crystalline lens, of perfect transparency, crossed double-convex, with the deepest curve inwards, marked B. The chamber between A and B is fitted with what is called the aqueous humour, not unlike water, which takes the form and acts the part of a meniscus lens ; the whole interior, behind B, is filled with a denser fluid, not unlike melted glass, and therefore called the vitreous humour. And then the whole interior of the choroid is 74 OPTICS WITHOUT MATHEMATICS. lined with another perfectly transparent coat, the retina (pronounced as rettind]^ an expansion of the optic nerve, which enters the eye at D, and runs back straight into the brain. And on this, in deep darkness, are formed the pictures of the outer world. But how are those pictures formed ? By a succession of lenses. The crys- talline lens and the two humours form the com- pound refracting apparatus by which all rays from external objects, not too close to the eye, are made to converge to a PF, or to conjugate foci, on the retina, which is spread out as a screen to receive them and transmit them to the brain ; and so exquisitely adapted to this pur- pose is its astonishingly intricate and complex structure, that scarcely anything is so minute as to escape its power. The image of a tree, with thousands upon thousands of leaves, may oc- cupy a space of not more than J inch upon the retina, and yet every leaf shall be perfectly de- picted 1 . But how this picture is communicated to the mind, or how seeing actually takes place, this is a mystery of which we know absolutely nothing. By the help of powerful microscopes we can penetrate a great way, far enough to be 1 Dr. Royston-Pigott states that under favourable circum- stances the eye can distinguish an object occupying TS^.TJOTJ^ f an inch on the retina. OPTICS WITHOUT MATHEMATICS. 75 astonished at what we are allowed to see ; but when we have reached a certain point, hitherto shalt thou come, but no further, the contact of mind and matter is one of the secret things that belong unto the Lord our God. But then I must tell you that besides the wonderful structure of this apparatus, there are several most curious adjustments that ensure its perfect use. There is the iris that I told you of; the coloured circle that is the great beauty of the eye. But it was not put there for beauty only ; though that was certainly one consideration, for the same object might have been answered by a much less sightly contrivance. The iris also by screening off the edges of the crystalline lens produces a sharper image. But now try this experiment with a candle in a dark room. Put your hand over any one's eye, so as to shut out all the light you can ; bring the candle close to it, and take away your hand suddenly, watching the eye all the time. You will see the pupil rapidly closing up, by the widening of that coloured ring. And what does that mean ? That the iris has the power of regulating the size of the pupil, so as to allow exactly the right quan- tity of light to enter the eye. Without this adjust- ment we should be dazzled in the sunshine, and purblind in twilight. The tincture of belladonna, 76 OPTICS WITHOUT MATHEMATICS. or deadly nightshade, has the strange property of destroying this power in the iris for the time ; oculists sometimes put a drop into the eye when they want to see the interior the pupil opens to its full size, and the dazzling and indis- tinctness are most disagreeable. Cat's eyes are a well-known instance of this adjustment ; closing up in a strong light, opening in the dark to see their prey. But in them the contraction is not circular, as in us, but up to a narrow slit. This arrangement is perfectly involuntary on our part. How perplexing and wearisome it would have been had these constantly recurring changes re- quired the effort of our own will, instead of being done for us in the utmost perfection ! even in this little matter we see how much we owe to the goodness of the Creator. And then there is another adjustment by which, through a slight movement of the crys- talline lens, we are able to bring the right focus on the retina at pleasure. You know that the PF and all the conjugate foci have each its own distance from the lens ; and that the images could not be distinct unless they were exactly in focus. The depth of the eye from front to back of course limits the distance of furthest dis- tinctness ; but inside that limit our focus can in some unknown way adjust itself. You can OPTICS WITHOUT MATHEMATICS. 77 easily try this. Hold up your finger near trie eye in front of a distant hill. You cannot see both, finger and hill, distinctly at once, but you can make either distinct in turn. You can choose which you wish to see distinctly, but you have nothing to do with the adjustment of focus which suits it. No one can explain this. And you see everything near you double, if you look (with both eyes) at a distant object ; and the reverse ; but another unconscious adjustment is given to us, by which we can turn the principal axes of both eyes at once on any object we please, so as to view it single. And thus again, because the pictures of near objects in the two eyes differ, each eye having a standpoint of its own, and that difference increases exactly with the nearness of the object, our mind judges of the distance in some unknown way, from the manner in which the two pictures are combined into one. This, by the way, is the principle of the stereo- scope^ in which two pictures, taken from different points, are so presented to the eyes that they are seen as one, and so, their different parts appear- ing at different distances as in nature, they convey the idea of solidity. I have been telling you about a perfect eye. But the eye is no more always perfect "than any other created object. Its lenses sometimes 78 OPTICS WITHOUT MATHEMATICS. refract too much, and sometimes too little, so that the foci fall short of the retina in the one case, or beyond it in the other, and then the rays either crossing before it, or not reaching their crossing, form little circles instead of points on the retina, which overlap one another and pro- duce an indistinct picture. The one defect where refraction is too strong is called short or near sight, common in youth, and said to be often caused or increased by a child's habit of learning with the book close to the face ; in this case very near objects only are seen distinctly, as their conjugate foci are long enough to reach the retina ; the opposite, long sight, in which very distant objects only are well seen, is the at- tendant of age. Cannot you suggest a remedy for either ? Nothing is easier. Place a suitable concave lens before the near-sighted eye, and a convex one before the other. In either case the total refraction will be altered, and a distinct image formed on the retina. And now you know all about eye-glasses and spectacles. Great comforts they are ; only if you should require such a thing, be sure to choose such as are rather too weak than too strong, lest you should increase the defect. Sometimes the crystalline lens (of which, by the way, you may see an excellent illustration in OPTICS WITHOUT MATHEMATICS. 79 the eye of a fish, a cod especially, only that it is a sphere instead of a crossed convex lens) becomes diseased and opaque, and makes the pupil look whitish, and of course produces blind- ness. This is called " cataract." It may be cured by "couching," that is, taking 'out the crystalline lens, or pushing it down out of the way ; a very deep convex spectacle lens is then required to supply the lost refraction, but with that the patient is enabled to see fairly well. Neither the first impression of light on the retina, nor its fading away, are quite instanta- neous. You cannot see the spokes of a moving carriage wheel : each has gone on to a fresh place before it has formed an image in the eye. You could not see the flight of a cannon-ball sideways, though you might endways. On the other hand, the lighted end of a stick will seem to produce a circle of fire if it is whirled round not slower than three times in a second, the duration depending, however, on the brightness. Hence we are not conscious of winking a beau- tiful provision for the constant cleansing of the cornea ; and this is the principle of the Cathe- rine wheel in fireworks, and of several ingenious toys, the zoetrope and others, in which the sight is bounded by a slit so that it cannot follow the moving image till a fresh one has been spun 80 OPTICS WITHOUT MATHEMATICS. into its place before the former has faded, so drawn as to produce a most curiously deceptive appearance of motion. Our time has I am sure not been ill spent in studying the wonders of the eye. But we must leave them now " half-told," and much less, and attend to one or two more points about refrac- tion before we quit that subject. A curious case may occur that we have thought nothing about. In passing out of a denser me- dium, suppose that the ratio of sines were to fail. What would happen then? It might occur in this way. We may suppose the angle of incidence to be gradually increased till the sine of the angle of refraction comes to nothing. Then what is the light to do ? It cannot pass through the surface, because the proportion must be kept : so it must go back. But it cannot go back by the law of refraction, for that requires two media, and here the light remains in the same. So re- flection takes the place of refraction, and it goes back at an equal angle ; and that is done in so perfect a way, none of the light escaping, that it is called Total Reflection. There is an experi- ment that will show this well, if carefully managed. Put water into a glass vessel, and place it above your eye, so that you can see the under surface of the water. Then if you set a OPTICS WITHOUT MATHEMATICS. 8 1 candle on one side of the vessel, and your eye on the other, so as to make a large angle with P on either side, and watch the under side of the water, you will find a position, by a little moving about, in which you will see a very perfect re- flection of the candle. This angle of reflection is sometimes called the critical angle. It has much to do with the brilliancy of jewels. Now for another point. You understood that a refracted ray goes on in a fresh straight line ; that is, supposing uniform density in the me- dium. But if the density differs in different parts, the ray will be no longer straight, for each different density will act like a fresh medium, and bend the ray into another direction. Our own atmosphere is a medium of this kind ; there is a general increase of density from its summit, some 40 miles high as commonly supposed, to the surface of the earth. You can see what effect this would have on the light of the heavenly bodies. If they were in the zenith (that is, right overhead), none at all, the rays coinciding with P. But in every direction to- G 8a OPTICS WITHOUT MATHEMATICS. wards the horizon there would be increasing refraction towards P, as the rays passed through increasingly dense layers of air to reach the sur- face ; and the effect would be to raise all the heavenly bodies, according to their height in the sky, above their true places. For let A be any place on the earth, B the atmosphere, C a star whose light is gradually refracted in a curve down to A : then, as we see by the last ray, the star will appear to be at c. The proportions are of course exaggerated in the diagram, but the effect is so considerable at the horizon as to raise the sun or moon rather more than their own breadth above their places. Now, do you see the result of this ? Why it must be that we shall see them quite above the horizon, when they are really not yet risen, or after they have set : and so every day in the year is at a mean 6 minutes at least longer owing to refraction even here, and much more if we were situated nearer the N. pole. OPTICS WITHOUT MATHEMATICS. 83 But besides this constant effect, there are per- petual variations of density from hot and cold currents mixing in the atmosphere which pro- duce irregular refraction. If ever you are al- lowed to use a large telescope, you will soon find this out. Many a brilliant night will prove a sad disappointment, from the fluttering and flaring that never will be still. And sometimes very strange appearances result from sudden local changes. On one occasion after a heavy shower following dry weather in India, the upper part of the Himalaya Mountains, where usually out of sight, started up from beneath the horizon. The whole of Dover Castle was seen by Dr. Vince from Ramsgate, August 6, 1806, on the nearer side of a hill which commonly conceals all but the tops of its four turrets. And in a yet more marvellous instance the atmo- sphere seems to have acted the part of a convex lens, for on July 36, 1798, the French coast, 40 or 50 miles distant, was seen from Hastings as if but a little way off, and the astonished fisher- men and sailors pointed out from the top of a hill the places they had been accustomed to visit, from Calais to St. Valery, and some said even to Dieppe. The mirage of the sandy deserts, where the thirsty traveller is tantalised by a vision of a lake in front of him which flies G 2 84 OPTICS WITHOUT MATHEMATICS. from his approach, is probably a case of total reflection of the cloudless sky from an intensely heated layer of air close to the burning sand. Deceptions of this kind, though less striking, may often be seen at the sea-side by those who know what to look for. When next you go there stand near the water's edge and look across it to some well-known distant object, an island, or cape coming down into the water, or shipping far away. You may very probably see, as I have repeatedly done in calm warm weather, singular distortions in the sea-horizon the " offing," as sailors call it outlines of land mis-shapen, or broken, or raised into the air, or perhaps the upper parts of vessels reflected upside down above the reality. It is now time for us to enter upon the most beautiful part of our subject, Colour. Very beautiful it is, and still more beautiful we should think it if we were not so used to it. But what would the world be without it? No charming tints of flowers, or gems, or landscapes, or sun- sets : one weary monotony of black, and white, and grey. And yet in that dull scene all the occupations of life might have gone on even as they do now. What can show more plainly the goodness of our Father in heaven than this loveliness of the habitation where He permits us OPTICS WITHOUT MATHEMATICS.- 85 to dwell! Well might the saintly Wulfstan l exclaim, " What must be the fair beauty of the Creator, Whose creatures are made so fair!" All the beauty of the world that does not de- pend on form or proportion is owing to Colour ; and colour is in many cases a subdivision' of Re- fraction. Reflection has nothing to do with it. It can and constantly does repeat it, but it does not cause it. That is the work among other causes of Refraction. But how is it done ? No doubt you will be anxious to know ; but it will involve a little retracing of our steps, a small amount of unlearning, which I am sorry for, but could not help. I could not have taught you in any other manner without adding greatly to your difficulties, and we can easily go back a little way. This then is what we have par- ticularly to notice. All through our refraction lessons we considered light to be simply white, or if the radiant was coloured, to keep that colour unaltered. We could not do otherwise at that time without introducing a great compli- cation. But now, our new lesson is to be this : that white light ceases to be white in the very act of refraction. Different parts of it are acted upon unequally; and it is split up at once, or 1 Bishop of Worcester at the time of the Norman Conquest, and holding that See from A.D. 1062 to 1095. 86 OPTICS WITHOUT MATHEMATICS. dispersed^ as we call it in Optics, into a variety of colours, literally, all the colours of the rain- bow. This dispersion, however, amounts to but a small proportion of the whole refraction ; so that we have not gone far wrong in learning re- fraction as if the light continued white; but it is time for us now to get that little wrong quite right. Let us think, then, what would be likely to show this dispersion into colour best. We should naturally try the simplest form of refrac- tion what we began with. You remember what that was a prism. And so thought Sir Isaac Newton ; and this was the way he went to work 1 . Through a little hole in a shutter he admitted a sunbeam into a dark room, and caught it on a prism. You can tell me what happened. Of course it was bent into an en- tirely fresh line : and if the edge of the prism were turned down, the ray would be bent up, which would be convenient, and might be caught on the wall, or a screen. But now can you tell me what happened further to the ray? This is what Newton learned, and what you have to learn. It was all broken up into beautiful colours, which, as the refraction was upwards only and not sideways, formed on the screen a long upright 1 About 1666. OPTICS WITHOUT MATHEMATICS. painted space, which is called the spectrum ; in this case the solar spectrum as being formed from the light of the sun. And he made out 7 colours in it, which, reckoning upwards from the least to the most refracted, he called Red, Orange, Yel- low, Green, Blue, Indigo, Violet. But there is no sharp division between these colours, every possible shade is there, melting into the next on each side, and we might as well say 70 as 7, if only we had 70 names for them. None of these can be decomposed any further, as you would find if you let it pass through a hole in the screen, and caught it on another prism. But if we were to put a second prism close behind the first, only turned the other way, or if we were to put a large convex lens in the beam of coloured rays, we shall find all these colours 88 OPTICS WITHOUT MATHEMATICS. would recombine into white light again, as at first. Now why was it necessary to use a little hole in this experiment ? Because, though the prism would decompose as large a beam as it could catch, yet a great part of the colours would overlap, and make white light again, and only the ends of the spectrum would be coloured ; so that the smaller the hole, the purer the spec- trum. But if we have a very small hole, we have very little light ; and to remedy this we can use a narrow slit, vertically no larger than the hole, but horizontally the full width of the prism ; this would act like a number of little holes side by side, and overlapping will be cut off, as the ray is not refracted sideways, but in a plane at right angles to the slit. Newton did not think of this improvement, which was devised by Wollaston in 1802, and has proved very effective. But now you will want to know how these colours are produced in the Rainbow. This is the principle of it. Every drop of rain refracts and disperses the little ray of light that enters it : this, again, suffers total reflection from the back of the drop, where the curve is too deep to let it pass through, and it is thus turned back, and on leaving the drop is refracted again back- wards and downwards towards the earth, forming OPTICS WITHOUT MATHEMATICS. 89 a diverging spectrum, that grows longer the fur- ther it goes, and by the time it reaches our eye, is so long in proportion to the size of our pupil, that we catch but one shade of colour out of the spectrum of one drop. But then the next drop above or below, making a slightly different angle with the sun and the eye, sends us the next shade of colour, and so, from a whole series of drops we get a whole series of colours, one from each ; and thus the top of the rainbow, where the action is ver- tical, is complete. But how as to the rest of the circle ? You have noticed that it was a cer- tain angle at the drop between the sun and your eye that sent you a certain colour. Now those angles would be formed just the same sideways as vertically, and in fact all round a circle, and therefore we see the same colours all round as much of the circle as lies above the earth ; and therefore again, because all the angles are equal, all rainbows are of the same size ; and we never see a mid-day rainbow in summer, for its top never rises above the ground ; and it is only a semicircle at sunrise or sunset ; and no two persons see the same rain- bow, because, though the angles are the same, 90 OPTICS WITHOUT MATHEMATICS. they must be formed at different drops to reach a fresh position ; and we see a fresh rainbow whenever we ourselves change our place. In the outer or secondary bow that we often see, the spectrum is formed by the rays that, entering the lower instead of the upper part of the drop, are reflected twice before they emerge, so that the colours are reversed. The green and violet fringes inside a bright bow are referred to "interference," which will be mentioned here- after. The bow, or " iris," is often seen in the spray of waterfalls and fountains, and dew-drops get their tints in the same way. The lunar rainbow is the same with the solar, but for ob- vious reasons much less commonly seen ; the moon is only bright enough to produce it during part of the month, and few persons are out all night in showery weather. Solar and lunar halos, parhelia, and other curious phenomena, which might be much more frequently seen if people looked for them, are due to refraction among minute crystals of ice in very high regions. But independently of its special interest as the token of the covenant, none can compare with the rainbow in its full development ; nor can we wonder at the words of the son of Sirach, "Look upon the rainbow, and praise Him that made it ; very beautiful it is in the brightness thereof : it OPTICS WITHOUT MATHEMATICS. 91 compasseth the heaven about with a glorious circle, and the hands of the Most High have bended it." We will now go back into our dark room to study the spectrum more at leisure. We per- ceive at once that the dispersion of light implies its unequal refraction, so that every colour has a refraction, or to speak in optical language, ref rangibility of its own, the red least and the violet most, so that the index of refraction must be given for the middle of the spectrum. And we have got to learn what we must take on trust, for the process that demonstrates it is rather troublesome that every colour has its own measure of vibration, its own distance be- tween the successive undulations, or as it is called, its own wave-length. When we were beginning you may remember that I gave you an easy average of 50,000 in an inch, for I could not talk about dispersion then ; but now we must be more particular, and take about 37,000 in an inch for the red, and 60,000 for the violet 1 , all intermediate colours having inter- mediate wave-lengths, with also corresponding velocities, from about 460 (red) to 730 (violet) billions of times per second. 1 Sir J. Herschel's numbers nearly. But they are differently given by others. 92 OPTICS WITHOUT MATHEMATICS. There is a good deal more to be known about the spectrum, a great deal more if we were to enter into minutiae. You could tell me that a prism might be made of anything transparent in that triangular form, even of a fluid if con- fined by thin glass plates ; but you could not tell me, what you would soon find, that prisms of different media would give different spectra ; some would be much longer than others, and though the order of the colours would be the same 1 , their proportions would differ ; the middle colour would not be the same in all kinds of glass, and no glass would expand the violet end as much as quartz, where it runs on in a kind of lavender grey, and with the electric light gives a spectrum 6 or 8 times as long as the common one from glass. The blue and violet rays have the strongest photographic action, so that those colours are unsuitable in the dress of the sitter ; and so are red and yellow, where the action is weakest ; hair of those colours does not come out well ; and a photographer employs red glass when he wants a light that will not spoil his work. The distribution of heat along the spectrum is rather a disputed matter, and does not belong to our subject ; in general it may be said that 1 With one exception, Ftuhsine, which transposes the two ends. OPTICS WITHOUT MATHEMATICS. 93 it belongs to the red much more than to the violet end, and extends beyond the red into the dark space. A beautiful modern instrument for studying the spectrum is called a spectroscope. Here we have first of all a slit, the breadth of which can be adjusted to the brightness of the light, then a convex lens concentrates the pencil on a prism, or set, or train of prisms to increase the dis- persion, and then an eyehole directs the sight, or a telescope magnifies the spectrum. This apparatus has led to very remarkable discoveries. The spectrum from a white hot solid body, or from many flames, is continuous, that is, without interruption from end to end. That from vapour or gas when incandescent, that is, heated so as to shine without burning, differs according to the material or " element," as chemists term it, every one of the 60 or 70 elements having a spectrum of its own ; these are sometimes bright narrow lines separated by darkness, some- times broad or fuzzy bands, but in all cases coloured according to their place in the spectrum; and these lines or bands are sometimes shown out black if there happens to be a brighter back- ground. And when we look at the solar spec- trum, we find it crossed from side to side by a multitude of hair lines, literally thousands in 94 OPTICS WITHOUT MATHEMATICS. large instruments, of different thickness and un- intelligible arrangement. Many of these can be matched in a specially constructed instrument with the lines given by terrestrial elements, and thus we are brought to the startling conclusion, that though we have no idea what makes the brilliant light of the body of the sun, it is closely surrounded by heated vapours belonging to ma- terials which we are familiar with, hydrogen, iron, magnesium, sodium, and other metals. And, stranger still, we find a corresponding though not identical composition in many stars also which are bright enough for examination. But we must not go on with this ; it belongs to that most wonderful science of Astronomy. I hope this little taste of it will lead you to study it as soon as you can 1 ; but we must finish our Optics first, and say something about Colour from an artist's point of view. It is a question not yet decided whether every colour is independent and unmixed throughout the spectrum, only melting imperceptibly into the adjoining tints, or whether several of them are not formed by mixture. Painters, who can com- pound any tint with red, yellow, and blue, which they consider " primary colours," of course favour 1 See Beckett's "Astronomy without Mathematics," on the list of the Society for Promoting Christian Knowledge. OPTICS WITHOUT MATHEMATICS. 95 this latter idea ; and every one supposes that yellow and blue form green ; yet against this it is urged that though they do so with paints, from a process of absorption which we have not yet studied, yet with coloured lights they do not, but form a greyish white. Red, green, and blue or violet have been advocated by others as prima- ries. You would be astonished to see how the spectroscope decomposes the most brilliant yel- low of flowers into red and green, and in fact leaves very little pure yellow to be found any- where. There is much to be said about it, and it would lead us too far. But we ought to know something about complementary (not complimen- tary) colours^ that is, such as are the complement or filling up of each other to make a perfect white. Thus, red by itself is complementary to the rest of the spectrum, where yellow and violet, being comparatively feeble and neutralising each other, the chief remainder, as complementary to red, is a bluish green ; blue in like manner is complemen- tary to red and yellow, forming orange, the green and violet compensating each other ; and violet is complementary to greenish yellow, because the red and blue are taken up in strengthening it. These complementary colours, being most oppo- site to each other, produce the strongest con- trasts, each heightening the other, and their 96 OPTICS WITHOUT MATHEMATICS. importance is well known in painting and dress. Accidental colours are of the same nature. If you take a piece of black paper and stick a white circle on it as large as a wafer, and cover it with a red wafer, which after gazing at it for a little while you take away, you will see a green spot in its place ; a broken pane in a green- glass window will give a strong pink hue to a white cloud seen through it ; and the full moon was seen by the astronomer Schmidt, when en- compassed by ruddy smoke on Vesuvius, of a bottle-green hue. The preponderance of yellow in gas- or candle-light causes most other colours to appear to disadvantage. Strange to say, many eyes are very defective in their perception of colour ; so many as to make a careful selection necessary among rail- way servants. Some few can see no colours at all ; others only blue and yellow ; some con- found red with green, and fall into all sorts of strange and absurd mistakes. But where rail- way signals are concerned, these become very serious. This colour-blindness is sometimes called " Daltonism," from the celebrated chemist of that name, who suffered from this defect. But leaving these matters to return to our original assertion, that refraction always dis- perses light into colour, and to the remarkable OPTICS WITHOUT MATHEMATICS. 97 instance of it given by the prism, you may ask whether it ought not to be found in the case of refraction through lenses also? We do find it, though less conspicuously, and the telescope maker wishes he did not find it, for it gives him a great deal of trouble. I spoke to you about the images in the foci of convex lenses as if they were colourless, or at least had only the colours of the objects they represented. But you can easily see how different the real case is. Take a convex lens with a great aperture or breadth in- proportion to its focus, a large reading-glass will do very well ; hold it up to the sun, and look at the circle of light a little inside and outside of the focus. You will find it strongly bordered with red in the one case and blue in the other. In fact the one pure simple image you have been thinking of, whether of the sun or anything else, does not exist. There is only a kind of compromise among a number of coloured images, each formed at its own distance from the lens, the red, as least refracted, going furthest ; the violet, for the opposite reason, drawing nearest to the lens, the others forming each its image between them ; and of course those that are not brought to a point in focus forming diffused fringes round the one which happens to be so. Now would not this be a grievous hindrance to H 98 OPTICS WITHOUT MATHEMATICS. telescopic vision, where all depends upon your magnifying an image as like as possible to the object itself? Of course it would. The moon, for instance, would be bordered with colours in such a way as to make it a very pretty picture for a child, but a most annoying object for an astronomer. And is there no way of remedy- ing this? The old telescope-makers tried the most obvious way, that of making the focal length of the object-glass^ or lens next the object, very great in proportion to its aperture; for thus, the angle of refraction being small, the amount of colour produced by it will be small also. But this was not only an imperfect but a very clumsy remedy. An image must be bright to bear much magnifying ; but you can- not have a bright image without a large lens ; and on this plan a large lens must bear an un- manageable focal length. If it were i inch and T Vths in diameter or aperture, they gave it 10 feet focus, and then it was expected to bear a power of 62 times ; if 3 inches and T 8 ^ths aper- ture, about 50 feet for a power of 140 ; and a 6-inch glass would have had more than 120 feet focus to magnify 216 times. But who ever heard of such telescopes ? How could they make tubes long enough ? how could they manage them ? Well, you cannot see such OPTICS WITHOUT MATHEMATICS. 99 things now, but you might 200 years ago ; mar- vellous things, as ingenious as unmanageable; braced up in strange ways to keep them firm, and mounted on long poles like masts of vessels. There are prints of such things remaining. , And when they got exceedingly long, they dispensed with tubes, which you can see have nothing to do with the principle of the telescope, and only serve to keep the lenses straight and rightly centred^ as it is called, and shut out extraneous light; and they hoisted the object-glass upon a mast, with cords to pull it up and down, and turn it in any direction, and they held the eye- lens or lenses, or eye-piece^ in the hand. We can only give them especial credit for discovering as much as they did. The celebrated Huygens, who died in 1695, had such an object-glass of 6 inches, and about 126 feet focus. Would you have had such a thing as a gift ? I imagine not. Dr. Derham, who had the care of it at one time, could not use it for want of a pole 100 feet or more high, which would have cost him 80 or 90. The inconvenience was so strongly felt that it led to the invention of the Reflecting Tele scope > which I did not mention when we were upon that subject, as you would not have then understood the action of the eye-lens in magnifying the image. A concave H 2, 100 OPTICS WITHOUT MATHEMATICS. mirror, or speculum as it is often called, will form an image better than a convex lens, as being free from colour. Why not use this? You will find you have got into a great diffi- culty. When you take your eye-lens to look at the image your own head comes directly in the way ; and how is this to be got rid of? There are several modes of doing it. Gregory, the first inventor of the reflecting telescope, de- vised this plan. He proposed to have a speculum of very hard white metal, taking a fine polish, with a hole in the middle, which of course did no harm except by wasting some of the light ; this he set at the end of a brass tube, in which, towards the other end, he placed a much smaller concave mirror with a much shorter focus, facing the larger one, so that the image of a distant object formed by the large mirror should be in one of its conjugate foci, and a second image be formed in the other focus in the hole of the large speculum, and be viewed by an eye-piece from behind, as here shown. In looking through it you see nothing of the small mirror because it is OPTICS WITHOUT MATHEMATICS. IO1 entirely out of focus, nothing but the image which it reflects from the large one 1 . This is not bright, much light being always lost by reflection ; but some of these instruments define very well, and Gregorian reflectors are still to be met with. Another construction was invented by Casse- grain, a Frenchman, which bears his name. The smaller speculum is here replaced by a convex one, which catches the . converging rays before they come to focus, and reflects them to the same place as in the Grego- = rian, to which some observers have preferred it. A very large one of this construction, with a con- cave speculum of 4 feet, was erected at Mel- bourne in 1869. A third kind, the best of all, was not only invented by Sir Isaac Newton, but executed with his own hands in 1671, and that relic is still in the Library of the Royal Society. Here the great speculum is not perforated, and the reflected rays are caught on a plane mirror near the mouth of the tube, so inclined as to turn 1 For the same reason nothing is seen of a fly, or even a finger, on or near the object-glass of a refracting telescope. Only a small loss of light is the result. 102 OPTICS WITHOUT MATHEMATICS. them out sideways at right angles through an opening in the tube, where they are viewed by an eye-piece. With this kind, the " Newtonian reflector," Sir W. Herschel made many of his discoveries 1 . But all these different reflectors went out of fashion when the achromatic tele- scope, which we are going to describe, came into use, though the Earl of Rosse, at a later date, constructed an enormous Newtonian of 6 feet aperture, the largest telescope in the world, but, owing to its length and weight, not the most generally useful. A far simpler method of disposing of the difficulty is to tilt the large mirror so as to throw the focus a little on one side, dispense with a second reflection, and view the image direct with the eye-piece, turning one's back to the object. This, which is called the " Front view," was pro- posed by Le Maire, and extensively employed 1 But not with the high powers sometimes absurdly ascribed to them. OPTICS WITHOUT MATHEMATICS. 103 by both the Herschels with their large telescopes. The image is bright, as the loss in the second reflection is avoided, but it is less perfect. It is never so perfect- as when in the axis. You must have noticed in your experiments with convex lenses in the sunshine, what strange distortion there is at the focus if the lens is held awry, and though comparatively trifling in these telescopes, it is enough to spoil their perfection. All these reflecting telescopes sprang out of the extreme unwieldiness of the long refractor, which was considered irremediable, owing to a mistake of Sir Isaac Newton's, who made as few mistakes as any man, but was wrong about this. He imagined that in all substances the disper- sion bears a fixed proportion to the refraction. But this is not the case. Two kinds of glass may be found, one of which, flint glass, of which wine-glasses are made, has a stronger disper- sion than the other, crown or plate glass, that is, of course, supposing they refract equally. So that if we were to combine two lenses, 104 OPTICS WITHOUT MATHEMATICS. one of each kind of glass, a convex and a con- cave, of equal foci, though of course there would be no refraction, and the rays would pass through unchanged in that respect, yet colour would be shown, because the dispersive powers would be unequal. Now what would be the converse of this ? Of course, that the dispersions of the two lenses might be equal, if the refractions were unequal ; that is, if the focal lengths were dif- ferent, and then the rays would undergo re- fraction and yet be colourless. We have only to combine a concave of naturally stronger dis- persion but longer focus, and therefore in this case dispersing less, with a convex of naturally weaker dispersion but shorter focus, and therefore ac- tually dispersing as much as its companion, and the thing is done. The shorter focus, as you ought to remember (see p. 57), will prevail, and the combination will act as a convex lens and form an image which will be without colour ; and thus we have got the achromatic tele scope , in which an object-glass of 6 inches aperture may have a focus of 7 instead of 120 feet, and be equally colourless, you may guess how much more manageable. This beautiful discovery is due to a gentleman named Hall, about 1733, and independently, though later, to Dollond, whose telescopes were long celebrated. And it OPTICS WITHOUT MATHEMATICS. 105 had another almost equally important advantage. I hinted (see p. 48) that spherical surfaces, such as grinding would produce, were not capable of forming a perfect focal image. You would not be able to understand the mathematical reason- ing, but you may easily satisfy yourselves of it by experiment. Take our old friend the short- focused reading-glass, and cut a piece of card to fit into its setting and cover it up, all except a small round hole of a quarter of an inch, which we must cut out of the centre, and a little ring half as broad, to be cut from the edge all round, leaving a few bridges of card here and there to keep it in the setting of the lens. If we hold this in sunshine or candlelight, and observe care- fully how the rays come to focus on another card, we shall find that the sharpest image formed by the open ring at the edge is rather nearer to the lens than that from the hole in the centre. In fact, if we can suppose the whole lens made up of narrow rings outside one another, every ring would have its own focus, just as I showed you that the colours have ; and the sharpness of the general image suffers in proportion. This is called the spherical error or aberration. With the same focal length it gets worse rapidly as the aperture is increased ; and therefore the old telescope-makers were obliged to reduce it, as 105 OPTICS WITHOUT MATHEMATICS. they did the chromatic error, or that arising from colour, by giving such unmanageable lengths to their foci. Fortunately, however, this error can be got rid of in a much more convenient way : the two lenses of our object-glass have opposite curves, convex and concave, and therefore their spherical aberrations lie in opposite directions, and may be made to neutralise each other almost completely, by a suitable arrangement of the curves on which they depend, and which, as you can easily see, may be varied without changing the focal length l . And thus we have made this beautiful inven- tion very perfect indeed. Yet not entirely so, and for a curious reason. The opposite dis- persions of the two lenses may be made exactly equal in length, but if so, the intermediate colours will not balance one another, because with different kinds of glass they do not occupy exactly the same position in the spectrum. And so when the lenses are combined, some of the colours will not come in with the rest ; and though the image will be far more free from colour than with the old refractor, there will still 1 How much this aberration depends on the position of the curve is readily shown by the difference in the image of the sun formed by a deep plano-convex lens such as is used in the eye- piece of a telescope according as its convex face is turned to or from the sun. OPTICS WITHOUT MATHEMATICS. 107 be a little uncorrected, forming what is termed a secondary spectrum^ which becomes disagreeable with a high power. From this cause, and from the enormous price of large pieces of perfect glass, there ha? latterly been a reaction in favour of the reflecting prin- ciple in a new form. The mirrors are made of glass, which is preferable in every way to metal, and coated on the front surface with a silver film, so thin that I do not know how to describe it ; and Newtonian reflectors thus constructed are not only cheaper than achromatics, but are very useful and valuable \ The largest yet con- structed is by Calver, at Mr. Common's obser- vatory : its diameter is 37 inches. Those worked by With, though smaller, are very perfect. The largest achromatic is at Vienna, by Grubb, of 29 inches aperture ; and there are several others nearly as large in America and England. I explained to you the principle of the Com- pound Microscope ; but I have now to add, what you could not have understood before, that the same corrections, both for colour and spherical aberration, have been very successfully applied to the object-glass of that instrument also, with the result of great increase of magnifying power. 1 The spherical error, mentioned in the note on p. 24, may be removed by a peculiar mode of polishing. 108 OPTICS WITHOUT MATHEMATICS. There is a great deal more to be learned about the construction and management of both the microscope and telescope ; but it lies rather out of our way. One piece of advice, however, I must give you and that is, never to lose an opportunity of using either. They are both wonderful inventions, and you will be as much astonished at the extreme minuteness of the most finished perfection revealed to us by the one, as at the grandeur and magnificence that will be displayed by the other. Galileo said of his telescope that it was invented through the illumination of the Divine favour. And truly he spoke well : and the inventor of the microscope might have made the like claim. And who can use either of them aright without coming to the conclusion of the Psalmist, O Lord, how mani- fold are Thy works, in wisdom hast Thou made them all? I think we have now gone pretty carefully over the first outlines of reflection and refraction ; but there are some other less ordinary properties of light of which you ought to have some idea, though we cannot go far into them without mathematical knowledge. One of these is Double Refraction^ which does not mean refrac- tion twice over, but in two paths at once. This happens with many kinds of crystals, especially OPTICS WITHOUT MATHEMATICS. IOQ Iceland spar, in which, owing to some peculiar arrangement of structure, a ray of incident light is split into two, not halved in size, but in bright- ness. One half follows the usual law which you have learned, and is therefore called the, ordinary ray ; the other is called the extraordinary, or as it had better be spelt, the extra- ordinary ray. These do not separate far from one another, and if the crystal is turned round, while the light falls on it at the same angle, the extra- ordinary ray will revolve round the other in the same direction ; so that if you hold a card behind the crystal, you will find upon it two specks of light, one turning round the other : and anything looked at through such a crystal, except in one direction in which there is no re- fraction, appears double. There are several kinds of crystals producing varieties of this curious phenomenon in some of them both rays are extra-ordinary, that is, neither obeys the ordinary law of refraction. In gases or fluids it is not met with. But there is no occasion to enter further into these minute details ; only you should know that if you happen to meet with a crystal through which you see things double, it is owing to this property of double refraction. Another curious condition of light must not be 110 OPTICS WITHOUT MATHEMATICS. left out, which is called Polarisation. It is far more common than the last ; in fact much of the light around us is more or less polarised without our being aware of it. The name seems to have been taken from magnetism or electricity; but it is not particularly suitable, and it is better not to think about poles, but to try to suppose, that is for our present purpose, that every ray of natural light consists of undulations crossing each other in every possible direction, and that when polarised they are all turned in two directions at right angles, something in this way and How this comes to pass is quite beyond our comprehension, and fortunately beside our pur- pose; but so it is that all polished bodies, except metals, polarise the light that they reflect, provided it is incident at a certain angle, called & polarising angle, which is different in different substances ; the amount of polarisation diminish- ing in proportion as that angle is departed from : and so again all doubly refracting crystals polar- ise their two rays, but always in planes at right OPTICS WITHOUT MATHEMATICS. Ill angles to each other, the plane of polarisation being that which passes through the original and the polarised ray. Now this being under- stood, that in certain cases, both of reflection and refraction, all the undulations can be turned two ways at right angles, it also happens, you will not ask me how, that there is something corresponding in the structure of many bodies that will only receive them in those ways : so that when polarised light falls on such surfaces it will either pass freely, or be stopped and extin- guished, according as the direction of its undu- lations agrees or disagrees with the optical structure of the body 1 , the extinction being most complete when these are at right angles, and more or less partial in intermediate positions. There are several contrivances to show this very curious effect, called Polariscopes ; the most simple is the Tourmaline, a doubly refracting crystal found chiefly in Ceylon, which would be best of all but for its being coloured. If two thin slices of this, cut with reference to its optical property, are so set close together that one may be turned round in front of the other, 1 We must always bear in mind that the direction of a ray, as it passes onward in its course, is quite a distinct thing from the direction of the undulations which form light, and are always at right angles to the path of the ray. 112 OPTICS WITHOUT MATHEMATICS. all the light will pass through if their optical structure lies the same way, or all be quenched if the second is turned so as to cross the first at right angles; a partial extinction taking place at intermediate positions. This is the case of ordinary light, which gets polarised in passing through the first plate, and then transmitted or stopped by the agreeing or disagreeing polar- ising power of the second : of course, if rays have been already polarised elsewhere, one slice of tourmaline that can be turned round will answer the same purpose. In such experiments we call the first crystal or surface the polarizer^ the second the analyser. The use of an ana- lyser will show how much polarised light sur- rounds us ; if we only get it at the right angle we shall find polarised light reflected by glass, water, polished furniture, varnished pictures, shining leaves, and many other things, but not metals, for a reason that it would be very diffi- cult for us to understand. Many media have the power of changing or " rotating " the plane of polarisation ; and thus differences are sometimes shown in their nature that would not otherwise be detected. An in- strument called a " saccharometer " is used to ascertain the genuineness of sugar ; the juice of the cane, and that of the grape, turning the OPTICS WITHOUT MATHEMATICS. 113 plane in opposite directions. A more surprising and mysterious fact was discovered by Faraday that the plane of polarisation is shifted by the influence of a powerful magnet. And now we have a few words to say as to the Interference of light. This gives rise to so many effects that we could not avoid all notice of it, though it is rather troublesome. To take an illustration from water, let us imagine one suc- cession of waves flowing along a straight channel, and another exactly similar succession running in a more roundabout course that brings them slanting into the straight channel somewhere in its length say half way what will be the condition of the united waves after they have passed the junction ? You will see at once that it depends on the condition of the waves of the two sets with respect to each other when they join. If the crests and hollows, or troughs as seamen call them, of each set should fall in the same places, they would run on twice as high and deep as before ; and this would happen even if the longer channel had delayed its set by one or any number of whole waves. But if the delay amounted to half a wave, or any odd number, such as 3, or 5, or 7, of half- waves, then they would meet in what are called opposite phases^ crest against trough and trough I 114 OPTICS WITHOUT MATHEMATICS. against crest, and the wave-motion would from that point be destroyed. This is what happens with light, on an infinitely minute scale. If two rays join after moving in paths of unequal length, or through media of different densities, then if the undulations they carry are in the same phase, crest matching crest, we have double brightness : if they are in precisely con- trary phases, there will be total darkness, or par- tial if the opposition is imperfect. This is the in- terference of two sets of waves, which often pro- duces curious and beautiful effects, on a minute scale. You might little suppose that one is the lovely colouring of a soap-bubble. Let us see how this happens. First, part of the light is re- flected from the outside, and this of course is white; but another part passes through to the inside, and there again some of it is reflected back through the bubble, and unites with the first reflected light and might be expected to double it. But now suppose the thickness of the bubble should retard it by half a wave what would follow ? An interference that would destroy instead of doubling the first reflection. But then we must remember that every colour has a separate wave-length, and therefore only one colour would be destroyed by one thick- ness; a fresh colour would be destroyed by OPTICS WITHOUT MATHEMATICS. 115 another thickness that would not interfere with the first ; and since the thickness varies every- where, there would be stoppings out of first one colour and then another all over the bubble. But when one colour is stopped out of white light all the rest come forward, the brightest first ; so where red has been quenched we shall see a bluish green, where blue has been stopped out we shall find orange; where violet is blocked we shall get yellow ; and so a complete display of beautiful tints, not arising from dispersion, but interference, will be spread over the bubble. And if we could measure the thicknesses we could get at the wave-lengths : and this can be done in another way, where there are two reflections and a film of air between, of ex- treme but varying thinness. Newton got such a film by laying the flat face of a plano- convex lens on the convex face of a lens having a very long curvature of 50 feet ; a set of regular coloured rings was thus formed, and he was able to measure and calculate the thick- nesses that gave them; and in this way the wave-lengths of different colours have been as- certained. Many other cases of beautiful colours where very thin films are concerned, or where many lines naturally exist or are artificially drawn I 2 Il6 OPTICS WITHOUT MATHEMATICS. extremely close together 1 , are due to the same principle of interference in one form or other; but we must not go too far into these details, nor can we describe the splendid tints and sym- metrical forms produced by interposing suitable crystals or other materials producing double refraction or interference, into an apparatus in which they are illuminated by polarised light. These experiments are often introduced into lectures with marvellous effect. Diffraction means a little sideway extension of the undulations, bordering the edges of the ray. This, in connection with interference, pro- duces in a dark room coloured fringes to shadows, and various other minute phaenomena. Irradiation is the apparent enlargement of a brilliant object on a dark background. It is probably due to the structure of the retina, and is uncertain in its amount, though frequently occurring. You may have noticed an instance in the moon when very young. The dark side, faintly illuminated by "earth-shine," appears part of a smaller sphere than the brilliant crescent, in the hollow of which it seems to lie. 1 According to Brewster, there may be 5,000 grooves in an inch of mother-of-pearl ; and beautiful spectra have been produced by ruling 30,000 in an inch of the minutest lines on glass or speculum- metal with the point of a diamond. "Iris ornaments," at one time fashionable, depended upon this principle. OPTICS WITHOUT MATHEMATICS. Ii; There are other curious facts about light, increasing the difficulty and the wonder of the study. To some living beings, and in some cases where nothing would have led us to expect it, the Creator has been pleased to give a strange power of originating light, even of considerable intensity; such as glow-worms ; fire-flies, the beauty of which in multitudes can hardly be imagined by those who have never witnessed it 1 ; some lichens also, it is said, in mines ; and wonders are told of the occasional luminosity of the sea ; butcher's meat also, and rotten wood, sometimes surprise the beholder in this way. This remarkable property is called Phosphor- escence^ a term which, however, we must not misunderstand as having any but a nominal con- nection with the substance called " phosphorus," the shining of which is of a very different nature slow combustion and chemical change. Some substances again many in a slight degree, and some crystals and diamonds among them have a power of absorbing light and reproducing it in the dark ; of which kind of phosphorescence " Balmain's luminous paint " is a well-known in- stance ; an invention which would be of great value if the light could be made more per- 1 There seems no doubt as to the singular fact that in certain very hot summers a few have been seen in England. Il8 OPTICS WITHOUT MATHEMATICS. manent. Some fluid media again, especially solutions of quinine or the infusion of horse- chestnut bark, possess a remarkable property called Fluorescence^ by which the light-waves of some parts of the spectrum are retarded, and the corresponding colours changed, green being in some cases turned to red, and the rays beyond the violet, usually too rapid for our sight, being slackened to a beautiful delicate blue. The chemical effect of long-continued exposure to light, in bleaching^ or altering many colours, ought not to be omitted. But we can only just advert to these curious points (of which the latter is of considerable importance to the artist), and pass on to the natural conclusion of our subject, the Absorption of light, when in consequence of the nature of the medium on which it is incident, or through which it passes, its little waves are more or less extinguished. This process is constantly going on all around us, and so extensively as to keep the balance even with the continued production of light. All bodies absorb some portion of these undu- lations. One-fifth is supposed to disappear even in a vertical passage through our atmo- sphere, and one-sixtieth only to survive trans- mission from the horizon. One-half is stated to be quenched by 7 feet of salt water, and the OPTICS WITHOUT MATHEMATICS. 119 bottom of deep oceans is as dark as night : and there is always loss in reflection excepting at the "critical angle." No lens transmits, no mirror returns, the whole. But absorption is always an unequal process, excepting in cases of pure whiteness. When it acts on transmitted light, it gives colour to the medium as we look through it, by stopping out some part or other of the spectrum : a curious instance of how far this may go is the combination of blue finger- glass and copper sulphate (blue vitriol); each detains the rays transmitted by the other, and the result is, if duly proportioned, entire dark- ness. And again as to reflected light ; you can easily understand that if the whole spectrum is absorbed, the result will be blackness, and the body will be invisible excepting from its contrast with others, or a slight reflection of white light from its outside : such is the case with soot, coal, or dried ink. But if one colour, or one set of colours only should be absorbed, the rest will be scattered in all directions by diffused reflection, and give the body the colour in which we see it. So that if you were asked the reason of the endless variety of colour all around us, you would answer that, with a few exceptions, such as the rainbow, dew-drops, or interference tints, it all arises from the decomposition of white 120 OPTICS WITHOUT MATHEMATICS. light by absorption. Why does a rose exhibit its beautiful hue ? Because its material reflects red, and absorbs the rest of the spectrum. Why do its leaves appear green ? Because they have absorbed all other coloured light : and so on, not only through the primary hues, whichever they may be, but all the mixed and " broken " tints, as artists call them, which surround us on every side. The spectroscope in fact shows us that very little natural colour is unmixed, or as pure as it may appear to the eye. It finds green even in the red of the geranium blossom, red in its greenest leaves. To this irregular absorption (as we may think it), this gradation and intermixture of hue, we owe the greater part of the lovely variety of every natural scene. And in the midst of this transcendent beauty, the light from which it all springs is, by some unknown process in the great laboratory, gradu- ally expiring and drawing towards ultimate ex- tinction, at least in that form of being. It is not lost ; nothing is lost ; its vibrations pass into heat, or other forms of motion or combina- tion, and are utilised in ways above our compre- hension ; but as Light it appears no more. We have now learned a little about Light. Very little, indeed, compared with all that has been investigated by mathematical research ; OPTICS WITHOUT MATHEMATICS. 121 but enough, it may be hoped, to answer an excellent purpose in awakening in us a greater interest than we had before, in wonders that lie on every side of us, but pass unnoticed because we see them every day. But they are none the less wonderful for that. And surely it well becomes God's children to exercise the under- standing He has given them in the study of His wonderful works, however mysterious even in their easiest lineaments, and utterly incompre- hensible in their further investigation. And this is only what might be expected as we draw nearer to " the hiding of His Power." In a sense not given to us to understand, He dwelleth in the light which no man can approach unto ; and in the highest, because a perfectly immaterial signification, He Himself is LIGHT. INDEX. Aberration, chromatic, 98,106. spherical, 24, 105. Absorption of light, 118. Analyser, 112. Angle of deviation, 46. Angles, 13-15. Astronomical telescope, 69. Atmospheric reflection, 34. refraction, 82. unusual, 83. Axis, principal, oblique, 25. Balmain's paint, 117. Blackness from absorption, 1 19. Bleaching effect, 118. Burning glass, 58. mirror, 31. Camera lucida, 61. obscura, 60. photographic, 61. Cataract, 79. Colour, 84. accidental, 96. complementary, 95. primary, 94. Conjugate foci, 26, 57. Critical angle, 81. Diffraction, 116. Dispersion, 86. Dollond, 104. Double refraction, 108. Elements, terrestrial, 93. Eye, 72. Eyepiece, 99. Fluorescence, 118. Focal length of lens, 55. mirror, 26. Focus, 24. principal, 26. virtual, 26. Galileo's telescope, 70, 108. Halos, parhelia, &c., 90. Herschel, Sir W., 102. Huygens, 5, 99. Interference, 113. Irradiation, 116. Irregular reflection, 34. refraction, 83. Kaleidoscope, 22. Lens, 48. Lenses, different kinds, 49-51. Looking-glass, 19. Magic lantern, 66. Magnetised light, 113. Magnifying, principle of, 61, 66. Microscope, compound, 66. 124 INDEX. Microscope, simple, 63. Mirage, 83. Mirrors, different kinds, 17, 23, 33- Near and long sight, 78. Newton's experiment, 86, 103. Object-glass, 98. Objects and images, 27, 29, 57. Opacity, transparency, 13. Phosphorescence, 117. Plane, meaning of, 16. of polarisation, 1 1 1. Polarisation, no. Polariscope, in. Polariser and analyser, 112. Polarising angle, 1 10. Prism, 40, 86, 92. Radiant point, 12. Ratio of sines, 39. Rays, different kinds, i a. Reflecting telescopes, 99. Reflection, 15. diffused, 34. total, 80. Refraction, 34. index of, 42. Refraction, double, 108. Rosse, Earl of, 102. Saccharometer, 112. Secondary spectrum, 107. Sines (of angles), 37. 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