Part 8

If the above explanation should still seem to require closer attention than the general reader may be disposed to give, the following, suggested by a friend of mine—a very skilful mathematician—will be found still simpler: Suppose a stream to flow quite uniformly, and that at one place on its banks an observer is stationed, while at another higher up a person throws corks into the water at regular intervals, say ten corks per minute; then these will float down and pass the other observer, wherever he may be, at the rate of ten per minute, _if_ the cork-thrower is at rest. But if he saunters either up-stream or down-stream, the corks will no longer float past the other at the exact rate of ten per minute. If the thrower is sauntering down-stream, then, between throwing any cork and the next, he has walked a certain way down, and the tenth cork, instead of having to travel the same distance as the first before reaching the observer, has a shorter distance to travel, and so reaches that observer sooner. Or in fact, which some may find easier to see, this cork will be nearer to the first cork than it would have been if the thrower had remained still. The corks will lie at equal distances from each other, but these equal distances will be less than they would have been if the observer had been at rest. If, on the contrary, the cork-thrower saunters up-stream, the corks will be somewhat further apart than if he had remained at rest. And supposing the observer to know beforehand that the corks would be thrown in at the rate of ten a minute, he would know, if they passed him at a greater rate than ten a minute (or, in other words, at a less distance from each other than the stream traversed in the tenth of a minute), that the cork-thrower was travelling down-stream or approaching him; whereas, if fewer than ten a minute passed him, he would know that the cork-thrower was travelling away from him, or up-stream. But also, if the cork-thrower were at rest, and the observer moved up-stream—that is, towards him—the corks would pass him at a greater rate than ten a minute; whereas, if the observer were travelling down-stream, or from the thrower, they would pass him at a slower rate. If both were moving, it is easily seen that if their movement brought them nearer together, the number of corks passing the observer per minute would be increased, whereas if their movements set them further apart, the number passing him per minute would be diminished.

These illustrations, derived from the motions of water, suffice in reality for our purpose. The waves which are emitted by luminous bodies in space travel onwards like the water-waves or the corks of the preceding illustrations. If the body which emits them is rapidly approaching us, the waves are set closer together or narrowed; whereas, if the body is receding, they are thrown further apart or broadened. And if we can in any way recognize such narrowing or broadening of the light-waves, we know just as certainly that the source of light is approaching us or receding from us (as the case may be) as our observer in the second illustration would know from the distance between the corks whether his friend, the cork-thrower, was drawing near to him or travelling away from him.

But it may be convenient to give another illustration, drawn from waves, which, like those of light, are not themselves discernible by our senses—I refer to those aerial waves of compression and rarefaction which produce what we call sound. These waves are not only in this respect better suited than water-waves to illustrate our subject, but also because they travel in all directions through aerial space, not merely along a surface. The waves which produce a certain note, that is, which excite in our minds, through the auditory nerve, the impression corresponding to a certain tone, have a definite length. So long as the observer, and a source of sound vibrating in one particular period, remain both in the same place, the note is unchanged in tone, though it may grow louder or fainter according as the vibrations increase or diminish in amplitude. But if the source of sound is approaching the hearer, the waves are thrown closer together and the sound is rendered more acute (the longer waves giving the deeper sound); and, on the other hand, if the source of sound is receding from the hearer, the waves are thrown further apart and the sound is rendered graver. The _rationale_ of these changes is precisely the same as that of the changes described in the preceding illustrations. It might, perhaps, appear that in so saying we were dismissing the illustration from sound, at least as an independent one, because we are explaining the illustration by preceding illustrations. But in reality, while there is absolutely nothing new to be said respecting the increase and diminution of distances (as between the waves and corks of the preceding illustration), the illustration from sound has the immense advantage of admitting readily of experimental tests. It is necessary only that the rate of approach or recession should bear an appreciable proportion to the rate at which sound travels. For waves are shortened or lengthened by approach or recession by an amount which bears to the entire length of the wave the same proportion which the rate of approach or recession bears to the rate of the wave’s advance. Now it is not very difficult to obtain rates of approach or recession fairly comparable with the velocity of sound—about 364 yards per second. An express train at full speed travels, let us say, about 1800 yards per minute, or 30 yards per second. Such a velocity would suffice to reduce all the sound-waves proceeding from a bell or whistle upon the engine, by about one-twelfth part, for an observer at rest on a station platform approached by the engine. On the contrary, after the engine had passed him, the sound-waves proceeding from the same bell or whistle would be lengthened by one-twelfth. The difference between the two tones would be almost exactly three semitones. If the hearer, instead of being on a platform, were in a train carried past the other at the same rate, the difference between the tone of the bell in approaching and its tone in receding would be about three tones. It would not be at all difficult so to arrange matters, that while two bells were sounding the same note—_Mi_, let us say—one bell on one engine the other on the other, a traveller by one should hear his own engine’s bell, the bell of the approaching engine, and the bell of the same engine receding, as the three notes—_Do_—_Mi_—_Sol_, whose wave-lengths are as the numbers 15, 12, and 10. We have here differences very easily to be recognized even by those who are not musicians. Every one who travels much by train must have noticed how the tone of a whistle changes as the engine sounding it travels past. The change is not quite sharp, but very rapid, because the other engine does not approach with a certain velocity up to a definite moment and then recede with the same velocity. It could only do this by rushing through the hearer, which would render the experiment theoretically more exact but practically unsatisfactory. As it rushes past instead of through him, there is a brief time during which the rate of approach is rapidly being reduced to nothing, followed by a similarly brief time during which the rate of recession gradually increases from nothing up to the actual rate of the engines’ velocities added together.[12] The change of tone may be thus illustrated:—

A B representing the sound of the approaching whistle, B C representing the rapid degradation of sound as the engine rushes close past the hearer, and C D representing the sound of the receding whistle. When a bell is sounded on the engine, as in America, the effect is better recognized, as I had repeated occasion to notice during my travels in that country. Probably this is because the tone of a bell is in any case much more clearly recognized than the tone of a railway whistle. The change of tone as a clanging bell is carried swiftly past (by the combined motions of both trains) is not at all of such a nature as to require close attention for its detection.

However, the apparent variation of sound produced by rapid approach or recession has been tested by exact experiments. On a railway uniting Utrecht and Maarsen “were placed,” the late Professor Nichol wrote, “at intervals of something upwards of a thousand yards, three groups of musicians, who remained motionless during the requisite period. Another musician on the railway sounded at intervals one uniform note; and its effects on the ears of the stationary musicians have been fully published. From these, certainly—from the recorded changes between grave and the more acute, and _vice versâ_,—confirming, even _numerically_, what the relative velocities might have enabled one to predict, it appears justifiable to conclude that the general theory is correct; and that the note of any sound may be greatly modified, if not wholly changed, by the velocity of the individual hearing it,” or, he should have added, by the velocity of the source of sound: perhaps more correct than either, is the statement that the note may be altered by the approach or recession of the source of sound, whether that be caused by the motion of the sounding body, or of the hearer himself, or of both.

It is difficult, indeed, to understand how doubt can exist in the mind of any one competent to form an opinion on the matter, though, as we shall presently see, some students of science and one or two mathematicians have raised doubts as to the validity of the reasoning by which it is shown that a change should occur. That the reasoning is sound cannot, in reality, be questioned, and after careful examination of the arguments urged against it by one or two mathematicians, I can form no other opinion than that these arguments amount really but to an expression of inability to understand the matter. This may seem astonishing, but is explained when we remember that some mathematicians, by devoting their attention too particularly to special departments, lose, to a surprising degree, the power of dealing with subjects (even mathematical ones) outside their department. Apart from the soundness of the reasoning, the facts are unmistakably in accordance with the conclusion to which the reasoning points. Yet some few still entertain doubts, a circumstance which may prove a source of consolation to any who find themselves unable to follow the reasoning on which the effects of approach and recession on wave-lengths depend. Let such remember, however, that experiment in the case of the aerial waves producing sound, accords perfectly with theory, and that the waves which produce light are perfectly analogous (so far as this particular point is concerned) with the waves producing sound.

Ordinary white light, and many kinds of coloured light, may be compared with _noise_—that is, with a multitude of intermixed sounds. But light of one pure colour may be compared to sound of one determinate note. As the aerial waves producing the effect of one definite tone are all of one length, so the ethereal waves producing light of one definite colour are all of one length. Therefore if we approach or recede from a source of light emitting such waves, effects will result corresponding with what has been described above for the case of water-waves and sound-waves. If we approach the source of light, or if it approaches us, the waves will be shortened; if we recede from it, or if it recedes from us, the waves will be lengthened. But the colour of light depends on its wave-length, precisely as the tone of sound depends on its wave-length. The waves producing red light are longer than those producing orange light, these are longer than the waves producing yellow light; and so the wave-lengths shorten down from yellow to green, thence to blue, to indigo, and finally to violet. Thus if a body shining in reality with a pure green colour, approached the observer with a velocity comparable with that of light, it would seem blue, indigo, or violet, according to the rate of approach; whereas if it rapidly receded, it would seem yellow, orange, or red, according to the rate of recession.

Unfortunately in one sense, though very fortunately in many much more important respects, the rates of motion among the celestial bodies are _not_ comparable with the velocity of light, but are always so much less as to be almost rest by comparison. The velocity of light is about 187,000 miles per second, or, according to the measures of the solar system at present in vogue (which will shortly have to give place to somewhat larger measures, the result of observations made upon the recent transit of Venus), about 185,000 miles per second. The swiftest celestial motion of which we have ever had direct evidence was that of the comet of the year 1843, which, at the time of its nearest approach to the sun, was travelling at the rate of about 350 miles per second. This, compared with the velocity of light, is as the motion of a person taking six steps a minute, each less than half a yard long, to the rush of the swiftest express train. No body within our solar system can travel faster than this, the motion of a body falling upon the sun from an infinite distance being only about 370 miles per second when it reaches his surface. And though swifter motions probably exist among the bodies travelling around more massive suns than ours, yet of such motions we can never become cognizant. All the motions taking place among the stars themselves would appear to be very much less in amount. The most swiftly moving sun seems to travel but at the rate of about 50 or 60 miles per second.

Now let us consider how far a motion of 100 miles per second might be expected to modify the colour of pure green light—selecting green as the middle colour of the spectrum. The waves producing green light are of such a length, that 47,000 of them scarcely equal in length a single inch. Draw on paper an inch and divide it carefully into ten equal parts, or take such parts from a well-divided rule; divide one of these tenths into ten equal parts, as nearly as the eye will permit you to judge; then one of these parts, or about half the thickness of an average pin, would contain 475 of the waves of pure green light. The same length would equal the length of 440 waves of pure yellow light, and of 511 waves of pure blue light. (The green, yellow, and blue, here spoken of, are understood to be of the precise colour of the middle of the green, yellow, and blue parts of the spectrum.) Thus the green waves must be increased in the proportion of 475 to 440 to give yellow light, or reduced in the proportion of 511 to 475 to give blue light. For the first purpose, the velocity of recession must bear to the velocity of light the proportion which 30 bears to 475, or must be equal to rather more than one-sixteenth part of the velocity of light—say 11,600 miles per second. For the second purpose, the velocity of approach must bear to the velocity of light the proportion which 36 bears to 475, or must be nearly equal to one-thirteenth part of the velocity of light—say 14,300 miles per second. But the motions of the stars and other celestial bodies, and also the motions of matter in the sun, and so forth, are very much less than these. Except in the case of one or two comets (and always dismissing from consideration the amazing apparent velocities with which comets’ tails _seem_ to be formed), we may take 100 miles per second as the extreme limit of velocity with which we have to deal, in considering the application of our theory to the motions of recession and approach of celestial bodies. Thus in the case of recession the greatest possible change of colour in pure green light would be equivalent to the difference between the medium green of the spectrum, and the colour 1-116th part of the way from medium green to medium yellow; and in the case of approach, the change would correspond to the difference between the medium green and the colour 1-143rd part of the way from medium green to medium blue. Let any one look at a spectrum of fair length, or even at a correctly tinted painting of the solar spectrum, and note how utterly unrecognizable to ordinary vision is the difference of tint for even the twentieth part of the distance between medium green and medium yellow on one side or medium blue on the other, and he will recognize how utterly hopeless it would be to attempt to appreciate the change of colour due to the approach or recession of a luminous body shining with pure green light and moving at the tremendous rate of 100 miles per second. It would be hopeless, even though we had the medium green colour and the changed colour, either towards yellow or towards blue, placed side by side for comparison—how much more when the changed colour would have to be compared with the observer’s recollection of the medium colour, as seen on some other occasion!

But this is the least important of the difficulties affecting the application of this method by noting change of colour, as Doppler originally proposed. Another difficulty, which seems somehow to have wholly escaped Doppler’s attention, renders the colour test altogether unavailable. We do not get _pure_ light from any of the celestial bodies except certain gaseous clouds or nebulæ. From every sun we get, as from our own sun, all the colours of the rainbow. There may be an excess of some colours and a deficiency of others in any star, so as to give the star a tint, or even a very decided colour. But even a blood-red star, or a deep-blue or violet star, does not shine with pure light, for the spectroscope shows that the star has other colours than those producing the prevailing tint, and it is only the great _excess_ of red rays (all kinds of red, too) or of blue rays (of all kinds), and so on, which makes the star appear red, or blue, and so on, to the eye. By far the greater number of stars or suns show all the colours of the rainbow nearly equally distributed, as in the case of our own sun. Now imagine for a moment a white sun, which had been at rest, to begin suddenly to approach us so rapidly (travelling more than 10,000 miles per second) that the red rays became orange, the orange became yellow, the yellow green, the green blue, the blue indigo, the indigo violet, while the violet waves became too short to affect the sense of sight. Then, _if that were all_, that sun, being deprived of the red part of its light, would shine with a slightly bluish tinge, owing to the relative superabundance of rays from the violet end of the spectrum. We should be able to recognize such a change, yet not nearly so distinctly as if that sun had been shining with a pure green light, and suddenly beginning to approach us at the enormous rate just mentioned, changed in colour to full blue. _Though_, if that sun were all the time approaching us at the enormous rate imagined, we should be quite unable to tell whether its slightly bluish tinge were due to such motion of approach or to some inherent blueness in the light emitted by the star. Similarly, if a white sun suddenly began to recede so rapidly that its violet rays were turned to indigo, the indigo to blue, and so on, the orange rays turning to red, and the red rays disappearing altogether, then, _if that were all_, its light would become slightly reddish, owing to the relative superabundance of light from the red end of the spectrum; and we might distinguish the change, yet not so readily as if a sun shining with pure green light began to recede at the same enormous rate, and so shone with pure yellow light. _Though_, if that sun were all the time receding at that enormous rate, we should be quite unable to tell whether its slightly reddish hue were due to such motion of recession or to some inherent redness in its own lustre. _But in neither case would that be all._ In the former, the red rays would indeed become orange; but the rays beyond the red, which produce no effect upon vision, would be converted into red rays, and fill up the part of the spectrum deserted by the rays originally red. In the latter, the violet rays would indeed become indigo; but the rays beyond the violet, ordinarily producing no visible effect, would be converted into violet rays, and fill up the part of the spectrum deserted by the rays originally violet. Thus, despite the enormous velocity of approach in one case and of recession in the other, there would be no change whatever in the colour of the sun in either case. All the colours of the rainbow would still be present in the sun’s light, and it would therefore still be a white sun.

Doppler’s method would thus fail utterly, even though the stars were travelling hither and thither with motions a hundred times greater than the greatest known stellar motions.

This objection to Doppler’s theory, as originally proposed, was considered by me in an article on “Coloured Suns” in _Fraser’s Magazine_ for January, 1868. His theory, indeed, was originally promulgated not as affording a means of measuring stellar motions, but as a way of accounting for the colours of double stars. It was thus presented by Professor Nichol, in a chapter of his “Architecture of the Heavens,” on this special subject:—“The rapid motion of light reaches indeed one of those numbers which reason owns, while imagination ceases to comprehend them; but it is also true that the swiftness with which certain individuals of the double stars sweep past their perihelias, or rather their periasters, is amazing; and in this matter of colours, it must be recollected that the question solely regards the difference between the velocities of the waves constituent of colours, at those different stellar positions. Still it is a bold—even a magnificent idea; and if it can be reconciled with the permanent colours of the multitude of stars surrounding us—stars which too are moving in great orbits with immense velocities—it may be hailed almost as a positive discovery. It must obtain confirmation, or otherwise, so soon as we can compare with certainty the observed colorific changes of separate systems with the known fluctuations of their orbital motions.”

That was written a quarter of a century ago, when spectroscopic analysis, as we now know it, had no existence. Accordingly, while the fatal objection to Doppler’s original theory is overlooked on the one hand, the means of applying the principle underlying the theory, in a much more exact manner than Doppler could have hoped for, is overlooked on the other. Both points are noted in the article above referred to, in the same paragraph. “We may dismiss,” I there stated, “the theory started some years ago by the French astronomer, M. Doppler.” But, I presently added, “It is quite clear that the effects of a motion rapid enough to produce such a change” (_i.e._ a change of tint in a pure colour) “would shift the position of the whole spectrum—and this change would be readily detected by a reference to the spectral lines.” This is true, even to the word “readily.” Velocities which would produce an appreciable change of tint would produce “readily” detectible changes in the position of the spectral lines; the velocities actually existing among the star-motions would produce changes in the position of these lines detectible only with extreme difficulty, or perhaps in the majority of instances not detectible at all.

It has been in this way that the spectroscopic method has actually been applied.