metry, so did Boole accomplish the marriage of logic and algebra.

Analogy in the Theory of Undulations.

There is no class of phenomena which more thoroughly illustrates alike the power and weakness of analogy than the waves which agitate every kind of medium. All waves, whatsoever be the matter through which they pass, obey the principles of rhythmical or harmonic motion, and the subject therefore presents a fine field for mathematical generalisation. Each kind of medium may allow of waves. peculiar in their conditions, so that it is a beautiful exercise in analogical reasoning to decide how, in making inferences from one kind of medium to another, we must make allowance for difference of circumstances. The waves of the ocean are large and visible, and there are the yet greater tidal waves which extend around the globe. From such palpable cases of rhythmical movement we pass to waves of sound, varying in length from about 32 feet to a small fraction of an inch. We have but to imagine, if we can, the fortieth octave of the middle C of a piano, and we reach the undulations of yellow light, the ultra-violet being about the forty-first octave. Thus we pass from the palpable and evident to that which is obscure, if not incomprehensible. Yet the same phenomena of reflection, interference, and refraction, which we find in some kinds of waves, may be expected to occur, mutatis mutandis, in other kinds.

From the great to the small, from the evident to the obscure, is not only the natural order of inference, but it is the historical order of discovery. The physical science of the Greek philosophers must have remained incomplete, and their theories groundless, because they did not understand the nature of undulations. Their systems were based upon the notion of movement of translation from place to place. Modern science tends to the opposite notion that all motion is alternating or rhythmical, energy flowing onwards but matter remaining comparatively fixed in position. Diogenes Laertius indeed correctly compared the propagation of sound with the spreading of waves on the surface of water when disturbed by a stone, and Vitruvius dis

played a more complete comprehension of the same analogy. It remained for Newton to create the theory of undulatory motion in showing by mathematical deductive reasoning that the particles of an elastic fluid by vibrating backwards and forwards, might carry a pulse or wave moving from the source of disturbance, while the disturbed particles return to their place of rest. He was even able to make a first approximation by theoretical calculation to the velocity of sound-waves in the atmosphere. His theory of sound formed a hardly less important epoch in science than his far more celebrated theory of gravitation. It opened the way to all the subsequent applications of mechanical principles to the insensible motion of molecules. He seems to have been, too, upon the brink of another application of the same principles which would have advanced science by a century of progress, and made him the undisputed founder of all the theories of matter. He expressed opinions at various times that light might be due to undulatory movements of a medium occupying space, and in one intensely interesting sentence remarks that colours are probably vibrations of different lengths, "much after the manner that, in the sense of hearing, nature makes use of aërial vibrations of several bignesses to generate sounds of divers tones, for the analogy of nature is to be observed." He correctly foresaw that red and yellow light would consist of the longer undulations, and blue and violet of the shorter, while white light would be composed of an indiscriminate mixture of waves of various lengths. Newton almost overcame the strongest apparent difficulty of the undulatory theory of light, namely, the propagation of light in straight lines. For he observed that though waves of sound bend round an obstacle to some extent, they do not do so in the same degree as water-waves.2 He had but to extend the analogy proportionally to light-waves, and not only would the difficulty have vanished, but the true theory of diffraction would have been open to him. Unfortunately he had a preconceived theory that rays of light are bent from and not towards the shadow of a body, a theory which for once he did not sufficiently compare with observation to detect 1 Birch, History of the Royal Society, vol. iii. p. 262, quoted by Young, Works, vol. i. p. 246..

2 Opticks, Query 28, 3rd edit. p. 337.

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its falsity. I am not aware, too, that Newton has, in any of his works, displayed an understanding of the phenomena of interference without which his notion of waves must have been imperfect.

While the general principles of undulatory motion will be the same in whatever medium the motion takes place, the circumstances may be excessively different. Between light travelling 186,000 miles per second and sound travelling in air only about 1,100 feet in the same time, or almost 900,000 times as slowly, we cannot expect a close outward resemblance. There are great differences, too, in the character of the vibrations. Gases scarcely admit of transverse vibration, so that sound travelling in air is a longitudinal wave, the particles of air moving backwards and forwards in the same line in which the wave moves onwards. Light, on the other hand, appears to consist entirely in the movement of points of force transversely to the direction of propagation of the ray. The light-wave is partially analogous to the bending of a rod or of a stretched cord agitated at one end. Now this bending motion may take place in any one of an infinite number of planes, and waves of which the planes are perpendicular to each other cannot interfere any more than two perpendicular forces can interfere. The complicated phenomena of polarised light arise out of this transverse character of the luminous wave, and we must not expect to meet analogous phenomena in atmospheric sound-waves. It is conceivable that in solids we might produce transverse sound undulations, in which phenomena of polarisation might be reproduced. But it would appear that even between transverse sound and lightwaves the analogy holds true rather of the principles of harmonic motion than the circumstances of the vibrating medium; from experiment and theory it is inferred that the plane of polarisation in plane polarised light is perpendicular to instead of being coincident with the direction of vibration, as it would be in the case of transverse sound undulations. If so the laws of elastic forces are essentially different in application to the luminiferous ether and to ordinary solid bodies.1

1 Rankine, Philosophical Transactions (1856), vol. cxlvi. p. 282.

Analogy in Astronomy.

We shall be much assisted in gaining a true appreciation of the value of analogy in its feebler degrees, by considering how much it has contributed to the progress of astronomical science. Our point of observation is so fixed with regard to the universe, and our means of examining distant bodies are so restricted, that we are necessarily guided by limited and apparently feeble resemblances. In many cases the result has been confirmed by subsequent direct evidence of the most forcible character.

While the scientific world was divided in opinion between the Copernican and Ptolemaic systems, it was analogy which furnished the most satisfactory argument. Galileo discovered, by the use of his new telescope, the four small satellites which circulate round Jupiter, and make a miniature planetary world. These four Medicean Stars, as they were called, were plainly seen to revolve round Jupiter in various periods, but approximately in one plane, and astronomers irresistibly inferred that what might happen on the smaller scale might also be found true of the greater planetary system. This discovery gave "the holding turn," as Herschel expressed it, to the opinions of mankind. Even Francis Bacon, who, little to the credit of his scientific sagacity, had previously opposed the Coper nican views, now became convinced, saying "We affirm th solisequium of Venus and Mercury; since it has been found by Galileo that Jupiter also has attendants." Nor did Huyghens think it superfluous to adopt the analogy as a valid argument.1 Even in an advanced stage of physical astronomy, the Jovian system has not lost its analogical interest; for the mutual perturbations of the four satellites pass through all their phases within a few centuries, and thus enable us to verify in a miniature case the principles. of stability, which Laplace established for the great planetary system. Oscillations or disturbances which in the motions of the planets appear to be secular, because their periods extend over millions of years, can be watched, in the case of Jupiter's satellites, through complete revolutions within the historical period of astronomy.2

1 Cosmotheoros (1699), p. 16.

2 Laplace, System of the World, vol. ii. p. 316.

In obtaining a knowledge of the stellar universe we must sometimes depend upon precarious analogies. We still hold upon this ground the opinion, entertained by Bruno as long ago as 1591, that the stars may be suns attended by planets like our earth. This is the most probable first assumption, and it is supported by spectrum observations, which show the similarity of light derived. from many stars with that of the sun. But at the same time we learn by the prism that there are nebulæ and stars in conditions widely different from anything known in our system. In the course of time the analogy may perhaps be restored to comparative completeness by the discovery of suns in various stages of nebulous condensation. history of the evolution of our own world may be traced back in bodies less developed, or traced forwards in systems. more advanced towards the dissipation of energy, and the extinction of life. As in a great workshop, we may perhaps see the material work of Creation as it has progressed through thousands of millions of years.

The

In speculations concerning the physical condition of the planets and their satellites, we depend upon analogies of a weak character. We may be said to know that the moon has mountains and valleys, plains and ridges, volcanoes and streams of lava, and, in spite of the absence of air and water, the rocky surface of the moon presents so many familiar appearances that we do not hesitate to compare them with the features of our globe. We infer with high probability that Mars has polar snow and an atmosphere absorbing blue rays like our own; Jupiter undoubtedly possesses a cloudy atmosphere, possibly not unlike a magnified copy of that surrounding the earth, but our tendency to adopt analogies receives a salutary correction in the recently discovered fact that the atmosphere of Uranus contains hydrogen.

Philosophers have not stopped at these comparatively safe inferences, but have speculated on the existence of living creatures in other planets. Huyghens remarked that as we infer by analogy from the dissected body of a dog to that of a pig and ox or other animal of the same general form, and as we expect to find the same viscera, the heart, stomach, lungs, intestines, &c., in corresponding positions, so when we notice the similarity of the planets