alone, aided by analogy, could never lead to their discovery. To divine these enigmas we must be guided by theoretical ideas founded on a true hypothesis. The theory of luminous vibrations presents this character, and these precious advantages; for to it we owe the discovery of optical laws the most complicated and most difficult to divine.'

Physicists who embraced the barren emission theory had nothing but their own native capacity and quickness of observation to rely upon. Fresnel having once seized the conditions of the true undulatory theory, as previously stated by Young, was enabled by the mere manipulation of his mathematical symbols to foresee many of the complicated phenomena of light. Who could possibly suppose, or even believe on the ground of mere common sense, that by stopping a large portion of the rays passing through a circular aperture, the illumination of a point upon a screen behind the aperture might be many times multiplied. Yet this paradoxical effect was predicted by Fresnel, and verified both by himself, and in a careful repetition of the experiment in later years, by Billet. Comparatively few persons even now are aware that in the very middle point of the shadow of an opaque circular disc is a point of light sensibly as bright as if no disc had been interposed. This startling fact was deduced from Fresnel's theory by Poisson, and was then verified experimentally by Arago. Airy, again, was led by pure theory to predict that Newton's rings would present a modified appearance if produced between a lens of glass and a plate of metal. This effect happened to have been observed fifteen years before by Arago, unknown to Airy; but another prediction of Airy, that there would be a further modification of the rings when made between two substances of very different refractive indices, was verified by subsequent trial with a diamond. A reversal of the rings takes place

when the space intervening between the plates is filled with a substance of intermediate refractive power, another phenomenon predicted by theory and verified by experiment as Sir John Herschel has described. There is hardly a limit to the number of other complicated effects of the interference of rays of light under different circumstances which might be deduced from the mathematical expressions, if it were worth while, or which, being previously observed can be explained, as in an interesting case observed by Sir John Herschel and explained by Airy P.

By a somewhat different effort of scientific foresight, Fresnel discovered that any solid transparent medium might be endowed with the power of double refraction by mere compression. For as he attributed the peculiar refracting power of crystals to the unequal elasticity in different directions, he inferred that unequal elasticity, if artificially produced, would give similar phenomena. With a powerful screw and a piece of glass, he then produced not only the colours due to double refraction, but the actual duplication of images. Thus, by a great scientific generalisation, are the apparently unique properties of Iceland spar shown to belong to all transparent substances under certain conditions 9.

All other predictions in optical science are, however, thrown into the shade by the theoretical discovery of conical refraction by the late Sir W. R. Hamilton, of Dublin. In investigating the passage of light through certain crystals, Hamilton found that Fresnel had slightly misinterpreted his own formulæ, and that, when rightly understood, they indicated a phenomenon of a kind never witnessed. A small ray of light sent into a crystal of arragonite in a particular direction, becomes spread out Airy's, Mathematical Tracts,' 3rd edit. p. 312. 4 Young's Works,' vol. i. p. 412.

into an infinite number of rays, which form a hollow cone within the crystal, and a hollow cylinder when emerging from the opposite side. In another case, a somewhat different, but equally strange, effect is produced. These phenomena are peculiarly interesting, because cones and cylinders of light are not produced in any other cases. They are, in fact, wholly opposed to all analogy, and constitute singular, or exceptional cases, of a kind which we shall afterwards have to consider more fully. Their very strangeness rendered them peculiarly fitted to test the truth of the theory by which they were discovered; and when Professor Lloyd, at Hamilton's request, succeeded, after considerable difficulty, in witnessing the new appearances, no further doubt could remain of the validity of the great theory of waves, which we owe to Huyghens, Young, and Fresnel1.

Predictions from the Theory of Undulations.

It is curious to reflect that the undulations of light, although so inconceivably rapid and small, admit of more accurate observation and measurement than the waves of any other medium. But so far as we can carry out exact experiments on other kinds of waves, we find the phenomena of interference repeated, and analogy gives considerable powers of prediction. Sir John Herschel was perhaps the first to suggest that two sounds might be made to destroy each other by interferences. For if onehalf of a wave travelling through a tube could be separated, and conducted by a somewhat longer passage, so as, on rejoining the other half, to be one-quarter of a vibra

r Lloyd's 'Wave Theory,' Part ii. pp. 52–58. Babbage, 'Ninth Bridgwater Treatise,' p. 104, quoting Lloyd, 'Trans. of the Royal Irish Academy,' vol. xvii. Clifton, Quarterly Journal of Pure and Applied Mathematics,' January, 1860.

8

* Encyclopædia Metropolitana,' art. Sound, p. 753.

tion behindhand, the two portions would exactly neutralise each other. This experiment has recently been performed with success by Quincke and Königt. The interference arising between the waves from the two prongs of a tuning-fork was also predicted by theory, and proved to exist by Weber; indeed it may be observed by merely turning round a vibrating fork close to the ear".

It is a plain result of the theory of sound that, if we move rapidly towards a sounding body, or if it move rapidly towards us, the pitch of the sound will be a little more acute; and, vice versa, when the relative motion is in the opposite direction, the pitch will be more grave. It arises from the less or greater intervals of time between the successive strokes of waves upon the auditory nerve, according as the ear moves towards or from the source of sound relatively speaking. This effect was predicted by theory, and afterwards verified by the experiments of M. Buys Ballot, on Dutch railways, and of Mr. Scott Russell, in England'. Whenever, indeed, one railway train passes another, on the locomotive of which the whistle is being sounded, the drop in the acuteness of the sound may be noticed at the moment of passing. This change gives the sound a peculiar howling character, which many persons must have noticed. I have calculated that, with two trains travelling thirty miles an hour, the effect would amount to rather more than half a tone, and it would often amount to a tone. A corresponding effect is produced in the case of light undulations, when the eye and the luminous body rapidly approach or recede from each other. It is shown by a slight change in the refrangibility of the rays of light, and a consequent change in the place of the lines of the spectrum, which has been made to give most important and unexpected information conTyndall's 'Sound,' p. 261. u Ibid. p. 273.

x Ibid. p. 78.

cerning the relative approach or recession of many stars as regards the earth.

Tides are vast waves, and were the earth's surface entirely covered by an ocean of uniform depth, they would admit of very exact theoretical investigation. The wholly irregular form of the several seas introduces unknown quantities and complexities with which theory cannot cope. Nevertheless, Whewell, observing that the tides of the German Ocean consist of interfering waves, which arrive partly round the north of Scotland and partly through the British Channel, was enabled to predict that at a point about midway between Lowestoft and Brill on the coast of Holland, in latitude 52° 27′ N, and longitude 3 h. 14 m. E, no tides would be found to exist. At that point the two waves would be of exactly the same amount, but in opposite phases, so as to neutralise each other. This assertion was verified by a surveying vessel of the British navy.

Prediction in other Sciences.

Generations, or even centuries, will probably elapse before mankind are in possession of a mathematical theory of the constitution of matter as complete and satisfactory as the theory of gravitation. Nevertheless, mathematrol physicists have in recent years acquired a fair hold of some of the simple relations of the physical forces to etter, and the proof is found in some remarkable antisin ons of curious phenomena which had never been Professor James Thomson deduced from Carvy of heat that the application of pressure would ting-point of ice. He even ventured to

t of this effect, and his statement was

Pse the Inductive Sciences,' vol. ii. p. 471.