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is called, is then perpendicular, and the ray goes through the water without deviation to the right or left. In other words, the ray in the air and the ray in the water form one continuous straight line. But the least deviation from the perpendicular causes the ray to be broken, or 'refracted,' at the point of incidence. What, then, is the law of refraction discovered by Snell? It is this, that no matter how the angle of incidence, and with it the angle of refraction, may vary, the relative magnitude of two lines, dependent on these angles, and called their sines, remains, for the same medium, perfectly unchanged. Measure, in other words, for various angles, each of these two lines with a scale, and divide the length of the longer one by that of the shorter; then, however the lines individually vary in length, the quotient yielded by this division remains absolutely the same. It is, in fact, what is called the index of refraction of the medium.

3

Science is an organic growth, and accurate measurements give coherence to the scientific organism. Were it not for the antecedent discovery of the law of sines, founded as it was on exact measurements, the rainbow could not have been explained. Again and again, moreover, the angular distance of the rainbow from the sun had been determined and found constant. In this divine remembrancer there was no variableness. A line drawn from the sun to the rainbow, and another drawn from the rainbow to the observer's eye, always enclosed an angle of 41°. Whence this steadfastness of position -this inflexible adherence to a particular angle? Newton gave to De Dominis 3 the credit of the answer; but we really owe it to the the genius of Descartes. He followed with his mind's eye of rays light impinging on a raindrop. He saw them in part reflected from the outside surface of the drop. He saw them refracted on entering the drop, reflected from its back, and again refracted on their emergence. Descartes was acquainted with the law of Snell, and taking up his pen he calculated, by means of that law, the whole course of the rays. He proved that the vast majority of them escaped from the drop as divergent rays, and, on this account, soon became so enfeebled as to produce no sensible effect upon the eye of an observer. At one particular angle, however—namely, the angle 41° aforesaid— they emerged in a practically parallel sheaf. In their union was strength, for it was this particular sheaf which carried the light of the 'primary' rainbow to the eye.

There is a certain form of emotion called intellectual pleasure, which may be excited by poetry, literature, nature, or art. But I doubt whether among the pleasures of the intellect there is any more pure and concentrated than that experienced by the scientific man

3 Archbishop of Spalatro, and Primate of Dalmatia. Fled to England about 1616; became a Protestant, and was made Dean of Windsor. Returned to Italy and resumed his Catholicism; but was handed over to the Inquisition, and died in prison (Poggendorff's Biographical Dictionary.)

when a difficulty which has challenged the human mind for ages melts before his eyes, and recrystallises as an illustration of natural law. This pleasure was doubtless experienced by Descartes when he succeeded in placing upon its true physical basis the most splendid meteor of our atmosphere. Descartes showed, moreover, that the secondary bow' was produced when the rays of light underwent two reflections within the drop, and two refractions at the points of incidence and emergence.

It is said that Descartes behaved ungenerously to Snell-that, though acquainted with the unpublished papers of the learned Dutchman, he failed to acknowledge his indebtedness. On this I will not dwell, for I notice on the part of the public a tendency, at all events in some cases, to emphasise such shortcomings. The temporary weakness of a great man is often taken as a sample of his whole character. The spot upon the sun usurps the place of his 'surpassing glory.' This is not unfrequent, but it is nevertheless unfair.

Descartes proved that according to the principles of refraction, a circular band of light must appear in the heavens exactly where the rainbow is seen. But how are the colours of the bow to be accounted for? Here his penetrative mind came to the very verge of the solution, but the limits of knowledge at the time barred his further progress. He connected the colours of the rainbow with those produced by a prism; but then these latter needed explanation just as much as the colours of the bow itself. The solution, indeed, was not possible until the composite nature of white light had been demonstrated by Newton. Applying the law of Snell to the different colours of the spectrum, Newton proved that the primary bow must consist of a series of concentric circular bands, the largest of which is red, and the smallest violet; while in the secondary bow these colours must be reversed. The main secret of the rainbow, if I may use such language, was thus revealed.

I have said that each colour of the rainbow is carried to the eye by a sheaf of approximately parallel rays. But what determines this parallelism? Here our real difficulties begin, but they are to be surmounted by attention. Let us endeavour to follow the course of the solar rays before and after they impinge upon a spherical drop of water. Take first of all the ray that passes through the centre of the drop. This particular ray strikes the back of the drop as a perpendicular, its reflected portion returning along its own course. Take another ray close to this central one and parallel to it for the sun's rays when they reach the earth are parallel. When this second ray enters the drop it is refracted; on reaching the back of the drop it is there reflected, being a second time refracted on its emergence from the drop. Here the incident and the emergent rays enclose a small angle with each other. Take again a third ray a little further from the central one than the last. The drop will act upon it as it acted upon its neighbour, the incident

and emergent rays enclosing in this instance a larger angle than before. As we retreat further from the central ray the enlargement of this angle continues up to a certain point, where it reaches a maximum, after which further retreat from the central ray diminishes the angle. Now, a maximum resembles the ridge of a hill, or a watershed, from which the land falls in a slope at each side. In the case before us the divergence of the rays when they quit the raindrop would be represented by the steepness of the slope. On the top of the watershed—that is to say, in the neighbourhood of our maximum —is a kind of summit level, where the slope for some distance almost disappears. But the disappearance of the slope indicates, in the case of our raindrop, the absence of divergence. Hence we find that at our maximum, and close to it, there issues from the drop a sheaf of rays which are nearly, if not quite, parallel to each other. These are the so-called 'effective rays' of the rainbow.4

Let me here point to a series of measurements which will illustrate the gradual augmentation of the deflection just referred to until it reaches its maximum, and its gradual diminution at the other side of the maximum. The measures correspond to a series of angles of incidence which augment by steps of ten degrees.

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The figures in the column i express these angles, while under d we have in each case the accompanying deviation, or the angle enclosed by the incident and emergent rays. It will be seen that as the angle i increases, the deviation also increases up to 42° 28', after which, although the angle of incidence goes on augmenting, the deviation becomes less. The maximum 42° 28′ corresponds to an incidence of 60°, but in reality at this point we have already passed, by a small quantity, the exact maximum, which occurs between 58° and 59°. Its amount is 42° 30'. This deviation corresponds to the red band of the rainbow. In a precisely similar manner the other colours rise to their maximum, and fall on passing beyond it; the maximum for the violet band being 40° 30'. The entire width of

There is, in fact, a bundle of rays near the maximum, which, when they enter the drop, are converged by refraction almost exactly to the same point at its back. If the convergence were quite exact, then the symmetry of the liquid sphere would cause the rays to quit the drop as they entered it—that is to say, perfectly parallel. But inasmuch as the convergence is not quite exact, the parallelism after emergence is only approximate. The emergent rays cut each other at extremely sharp angles, thus forming a 'caustic' which has for its asymptote the ray of maximum deviation. In the secondary bow we have to deal with a minimum, instead of a maximum, the crossing of the incident and emergent rays producing the observed reversal of the colours. (See Engel and Shellbach's diagrams of the rainbow at the end of this article.)

the primary rainbow is therefore 2°, part of this width being due to the angular magnitude of the sun.

5

We have thus revealed to us the geometric construction of the rainbow. But though the step here taken by Descartes and Newton was a great one, it left the theory of the bow incomplete. Within the rainbow proper, in certain conditions of the atmosphere, are seen a series of richly-coloured zones, which were not explained by either Descartes or Newton. They are said to have been first described by Mariotte, and they long challenged explanation. At this point our difficulties thicken, but, as before, they are to be overcome by attention. It belongs to the very essence of a maximum, approached continuously on both sides, that on the two sides of it pairs of equal value may be found. The maximum density of water, for example, is 39° Fahrenheit. Its density when 5° colder, and when 5° warmer, than this maximum is the same. So also with regard to the slopes of our watershed. A series of pairs of points of the same elevation can be found upon the two sides of the ridge; and, in the case of the rainbow, on the two sides of the maximum deviation we have a succession of pairs of rays having the same deflection. Such rays travel along the same line, and add their forces together after they quit the drop. But, light, thus reinforced by the coalescence of non-divergent rays, ought to reach the eye. It does so; and were light what it was once supposed to be-a flight of minute particles sent by luminous bodies through space-then these pairs of equally deflected rays would diffuse brightness over a large portion of the area within the primary bow. But inasmuch as light consists of waves and not of particles, the principle of interference comes into play, in virtue of which waves can alternately reinforce and destroy each other. Were the distance passed over, by the two corresponding rays within the drop, the same, they would emerge as they entered. But in no case are the distances the same. The consequence is that when the rays emerge from the drop they are in a condition either to support or to destroy each other. By such alternate reinforcement and destruction, which occur at different places for different colours, the coloured zones are produced within the primary bow. They are called 'supernumerary bows,' and are seen, not only within the primary but sometimes also outside the secondary bow. The condition requisite for their production is, that the drops which constitute the shower shall all be of nearly the same size. When the drops are of different sizes, we have a confused superposition of the different colours, an approximation to white light being the consequence. This second step in the explanation of the rainbow was taken by a man the quality of whose genius resembled that of Descartes or Newton, and who eighty-two years ago was appointed Professor of Natural Philosophy in the Royal

5 Prior of St. Martin-sous-Beaune, near Dijon. Member of the French Academy of Sciences. Died in Paris, May 1684.

Institution of Great Britain. I refer, of course, to the illustrious Thomas Young.6

But our task is not, even now, complete. The finishing touch to the explanation of the rainbow was given by our last, eminent, Astronomer Royal, Sir George Airy. Bringing the knowledge possessed by the founders of the undulatory theory, and that gained by subsequent workers, to bear upon the question, Sir George Airy showed that, though Young's general principles were unassailable, his calculations were sometimes wide of the mark. It was proved by Airy that the curve of maximum illumination in the rainbow does not quite coincide with the geometric curve of Descartes and Newton. He also extended our knowledge of the supernumerary bows, and corrected the positions which Young had assigned to them. Finally, Professor Miller, of Cambridge, and Dr. Galle, of Berlin, illustrated by careful measurements with the theodolite the agreement which exists between the theory of Airy and the facts of observation. Thus, from Descartes to Airy, the intellectual force expended in the elucidation of the rainbow, though broken up into distinct personalties, might be regarded as that of an individual artist, engaged throughout this time in lovingly contemplating, revising, and perfecting his work.

We have thus cleared the ground for the series of experiments which constitute the subject of this discourse. During our brief residence in the Alps this year, we were favoured with some weather of matchless perfection; but we had also our share of foggy and drizzly weather. On the night of the 22nd of September, the atmosphere was especially dark and thick. At 9 P.M. I opened a door at the end of a passage and looked out into the gloom. Behind me hung a small lamp, by which the shadow of my body was cast upon the fog. Such a shadow I had often seen, but in the present case it was accompanied by an appearance which I had not previously seen. Swept through the darkness round the shadow, and far beyond, not only its boundary, but also beyond that of the illuminated fog, was a pale, white, luminous circle, complete except at the point where it was cut through by the shadow. As I walked out into the fog, this curious halo went in advance of me. Had not my demerits been so well known to me, I might have accepted the phenomenon as an evidence of canonisation. Benvenuto Cellini saw something of the kind surrounding his shadow, and ascribed it forthwith to supernatural favour. I varied the position and intensity of the lamp, and found even a candle sufficient to render the luminous band visible. With two crossed laths I roughly measured the angle subtended by the radius of the circle, and found it to be practically the angle which had riveted the attention of Descartes-namely, 41°. This and other

• Young's Works, edited by Peacock, vol. i. pp. 185, 293, 357.

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