CHAPTER XXIV.

EMPIRICAL KNOWLEDGE, EXPLANATION, AND PREDICTION.

Inductive investigation, as we have seen, consists in the union of hypothesis and experiment, deductive reasoning being the link by which experimental results are made to confirm or confute the hypothesis. Now when we consider this relation between hypothesis and experiment it is obvious that we may classify our knowledge under four heads.

(1) We may be acquainted with facts which have not yet been brought into accordance with any hypothesis. Such facts constitute what is called *Empirical Knowledge*.

(2) Another extensive portion of our knowledge consists of facts which having been first observed empirically, have afterwards been brought into accordance with other facts by an hypothesis concerning the general laws applying to them. This portion of our knowledge may be said to be *explained*, *reasoned*, or *generalised*.

(3) In the third place comes the collection of facts, minor in number, but most important as regards their scientific interest, which have been anticipated by theory and afterwards verified by experiment.

(4) Lastly, there exists knowledge which is accepted solely on the ground of theory, and is incapable of experimental confirmation, at least with the instrumental means in our possession.

It is a work of much interest to compare and illustrate the relative extent and value of these four groups of knowledge. We shall observe that as a general rule a great branch of science originates in facts observed accidentally, or without distinct consciousness of what is to be expected. As a science progresses, its power of foresight rapidly increases, until the mathematician in his library acquires the power of anticipating nature, and predicting what will happen in circumstances which the eye of man has never examined.

*Empirical Knowledge.*

By empirical knowledge we mean such as is derived directly from the examination of detached facts, and rests entirely on those facts, without corroboration from other branches of knowledge. It is contrasted with generalised and theoretical knowledge, which embraces many series of facts under a few comprehensive principles, so that each series serves to throw light upon each other series of facts. Just as, in the map of a half-explored country, we see detached bits of rivers, isolated mountains, and undefined plains, not connected into any complete plan, so a new branch of knowledge consists of groups of facts, each group standing apart, so as not to allow us to reason from one to another.

Before the time of Descartes, and Newton, and Huyghens, there was much empirical knowledge of the phenomena of light. The rainbow had always struck the attention of the most careless observers, and there was no difficulty in perceiving that its conditions of occurrence consisted in rays of the sun shining upon falling drops of rain. It was impossible to overlook the resemblance of the ordinary rainbow to the comparatively rare lunar rainbow, to the bow which appears upon the spray of a waterfall, or even upon beads of dew suspended on grass and spiders’ webs. In all these cases the uniform conditions are rays of light and round drops of water. Roger Bacon had noticed these conditions, as well as the analogy of the rainbow colours to those produced by crystals.[437] But the knowledge was empirical until Descartes and Newton showed how the phenomena were connected with facts concerning the refraction of light.

[437] *Opus Majus.* Edit. 1733. Cap. x. p. 460.

There can be no better instance of an empirical truth than that detected by Newton concerning the high refractive powers of combustible substances. Newton’s chemical notions were almost as vague as those prevalent in his day, but he observed that certain “fat, sulphureous, unctuous bodies,” as he calls them, such as camphor, oils spirit of turpentine, amber, &c., have refractive powers two or three times greater than might be anticipated from their densities.[438] The enormous refractive index of diamond, led him with great sagacity to regard this substance as of the same unctuous or inflammable nature, so that he may be regarded as predicting the combustibility of the diamond, afterwards demonstrated by the Florentine Academicians in 1694. Brewster having entered into a long investigation of the refractive powers of different substances, confirmed Newton’s assertions, and found that the three elementary combustible substances, diamond, phosphorus, and sulphur, have, in comparison with their densities, by far the highest known refractive indices,[439] and there are only a few substances, such as chromate of lead or glass of antimony, which exceed them in absolute power of refraction. The oils and hydrocarbons generally possess excessive indices. But all this knowledge remains to the present day purely empirical, no connection having been pointed out between this coincidence of inflammability and high refractive power, with other laws of chemistry or optics. It is worth notice, as pointed out by Brewster, that if Newton had argued concerning two minerals, Greenockite and Octahedrite, as he did concerning diamond, his predictions would have proved false, showing sufficiently that he did not make any sure induction on the subject. In the present day, the relation of the refractive index to the density and atomic weight of a substance is becoming a matter of theory; yet there remain specific differences of refracting power known only on empirical grounds, and it is curious that in hydrogen an abnormally high refractive power has been found to be joined to inflammability.

[438] Newton’s *Opticks*. Third edit. p. 249.

[439] Brewster. *Treatise on New Philosophical Instruments*, p. 266, &c.

The science of chemistry, however much its theory may have progressed, still presents us with a vast body of empirical knowledge. Not only is it as yet hopeless to attempt to account for the particular group of qualities belonging to each element, but there are multitudes of particular facts of which no further account can be given. Why should the sulphides of many metals be intensely black? Why should a slight amount of phosphoric acid have so great a power of interference with the crystallisation of vanadic acid?[440] Why should the compound silicates of alkalies and alkaline metals be transparent? Why should gold be so highly ductile, and gold and silver the only two sensibly translucent metals? Why should sulphur be capable of so many peculiar changes into allotropic modifications?

[440] Roscoe, Bakerian Lecture, *Philosophical Transactions* (1868), vol. clviii. p. 6.

There are whole branches of chemical knowledge which are mere collections of disconnected facts. The properties of alloys are often remarkable; but no laws have yet been detected, and the laws of combining proportions seem to have no clear application.[441] Not the slightest explanation can be given of the wonderful variations of the qualities of iron, according as it contains more or less carbon and silicon, nay, even the facts of the case are often involved in uncertainty. Why, again, should the properties of steel be remarkably affected by the presence of a little tungsten or manganese? All that was determined by Matthiessen concerning the conducting powers of copper, was of a purely empirical character.[442] Many animal substances cannot be shown to obey the laws of combining proportions. Thus for the most part chemistry is yet an empirical science occupied with the registration of immense numbers of disconnected facts, which may at some future time become the basis of a greatly extended theory.

[441] *Life of Faraday*, vol. ii. p. 104.

[442] Watts, *Dictionary of Chemistry*, vol. ii, p. 39, &c.

We must not indeed suppose that any science will ever entirely cease to be empirical. Multitudes of phenomena have been explained by the undulatory theory of light; but there yet remain many facts to be treated. The natural colours of bodies and the rays given off by them when heated, are unexplained, and yield few empirical coincidences. The theory of electricity is partially understood, but the conditions of the production of frictional electricity defy explanation, although they have been studied for two centuries. I shall subsequently point out that even the establishment of a wide and true law of nature is but the starting-point for the discovery of exceptions and divergences giving a new scope to empirical discovery.

There is probably no science, I have said, which is entirely free from empirical and unexplained facts. Logic approaches most nearly to this position, as it is merely a deductive development of the laws of thought and the principle of substitution. Yet some of the facts established in the investigation of the inverse logical problem may be considered empirical. That a proposition of the form A = BC ꖌ *b c* possesses the least number of distinct logical variations, and the greatest number of logical equivalents of the same form among propositions involving three classes (p. 141), is a case in point. So also is the fact discovered by Professor Clifford that in regard to statements involving four classes, there is only one example of two dissimilar statements having the same distances (p. 144). Mathematical science often yields empirical truths. Why, for instance, should the value of π, when expressed to a great number of figures, contain the digit 7 much less frequently than any other digit?[443] Even geometry may allow of empirical truths, when the matter does not involve quantities of space, but numerical results and the positive or negative character of quantities, as in De Morgan’s theorem concerning negative areas.

[443] De Morgan’s *Budget of Paradoxes*, p. 291.

*Accidental Discovery.*

There are not a few cases where almost pure accident has determined the moment when a new branch of knowledge was to be created. The laws of the structure of crystals were not discovered until Haüy happened to drop a beautiful crystal of calc-spar upon a stone pavement. His momentary regret at destroying a choice specimen was quickly removed when, in attempting to join the fragments together, he observed regular geometrical faces, which did not correspond with the external facets of the crystals. A great many more crystals were soon broken intentionally, to observe the planes of cleavage, and the discovery of the internal structure of crystalline substances was the result. Here we see how much more was due to the reasoning power of the philosopher, than to an accident which must often have happened to other persons.

In a similar manner, a fortuitous occurrence led Malus to discover the polarisation of light by reflection. The phenomena of double refraction had been long known, and when engaged in Paris in 1808, in investigating the character of light thus polarised, Malus chanced to look through a double refracting prism at the light of the setting sun, reflected from the windows of the Luxembourg Palace. In turning the prism round, he was surprised to find that the ordinary image disappeared at two opposite positions of the prism. He remarked that the reflected light behaved like light which had been polarised by passing through another prism. He was induced to test the character of light reflected under other circumstances, and it was eventually proved that polarisation is invariably connected with reflection. Some of the general laws of optics, previously unsuspected, were thus discovered by pure accident. In the history of electricity, accident has had a large part. For centuries some of the more common effects of magnetism and of frictional electricity had presented themselves as unaccountable deviations from the ordinary course of Nature. Accident must have first directed attention to such phenomena, but how few of those who witnessed them had any conception of the all-pervading character of the power manifested. The very existence of galvanism, or electricity of low tension, was unsuspected until Galvani accidentally touched the leg of a frog with pieces of metal. The decomposition of water by voltaic electricity also was accidentally discovered by Nicholson in 1801, and Davy speaks of this discovery as the foundation of all that had since been done in electro-chemical science.

It is otherwise with the discovery of electro-magnetism. Oersted, in common with many others, had suspected the existence of some relation between the magnet and electricity, and he appears to have tried to detect its exact nature. Once, as we are told by Hansteen, he had employed a strong galvanic battery during a lecture, and at the close it occurred to him to try the effect of placing the conducting wire parallel to a magnetic needle, instead of at right angles, as he had previously done. The needle immediately moved and took up a position nearly at right angles to the wire; he inverted the direction of the current, and the needle deviated in a contrary direction. The great discovery was made, and if by accident, it was such an accident as happens, as Lagrange remarked of Newton, only to those who deserve it.[444] There was, in fact, nothing accidental, except that, as in all totally new discoveries, Oersted did not know what to look for. He could not infer from previous knowledge the nature of the relation, and it was only repeated trial in different modes which could lead him to the right combination. High and happy powers of inference, and not accident, subsequently led Faraday to reverse the process, and to show that the motion of the magnet would occasion an electric current in the wire.

[444] *Life of Faraday*, vol. ii p. 396.

Sufficient investigation would probably show that almost every branch of art and science had an accidental beginning. In historical times almost every important new instrument as the telescope, the microscope, or the compass, was probably suggested by some accidental occurrence. In pre-historic times the germs of the arts must have arisen still more exclusively in the same way. Cultivation of plants probably arose, in Mr. Darwin’s opinion, from some such accident as the seeds of a fruit falling upon a heap of refuse, and producing an unusually fine variety. Even the use of fire must, some time or other, have been discovered in an accidental manner.

With the progress of a branch of science, the element of chance becomes much reduced. Not only are laws discovered which enable results to be predicted, as we shall see, but the systematic examination of phenomena and substances often leads to discoveries which can in no sense be said to be accidental. It has been asserted that the anæsthetic properties of chloroform were disclosed by a little dog smelling at a saucerful of the liquid in a chemist’s shop in Linlithgow, the singular effects upon the dog being reported to Simpson, who turned the incident to good account. This story, however, has been shown to be a fabrication, the fact being that Simpson had for many years been endeavouring to discover a better anæsthetic than those previously employed, and that he tested the properties of chloroform, among other substances, at the suggestion of Waldie, a Liverpool chemist. The valuable powers of chloral hydrate have since been discovered in a like manner, and systematic inquiries are continually being made into the therapeutic or economic values of new chemical compounds.

If we must attempt to draw a conclusion concerning the part which chance plays in scientific discovery, it must be allowed that it more or less affects the success of all inductive investigation, but becomes less important with the progress of science. Accident may bring a new and valuable combination to the notice of some person who had never expressly searched for a discovery of the kind, and the probabilities are certainly in favour of a discovery being occasionally made in this manner. But the greater the tact and industry with which a physicist applies himself to the study of nature, the greater is the probability that he will meet with fortunate accidents, and will turn them to good account. Thus it comes to pass that, in the refined investigations of the present day, genius united to extensive knowledge, cultivated powers, and indomitable industry, constitute the characteristics of the successful discoverer.

*Empirical Observations subsequently Explained.*

The second great portion of scientific knowledge consists of facts which have been first learnt in a purely empirical manner, but have afterwards been shown to follow from some law of nature, that is, from some highly probable hypothesis. Facts are said to be explained when they are thus brought into harmony with other facts, or bodies of general knowledge. There are few words more familiarly used in scientific phraseology than this word *explanation*, and it is necessary to decide exactly what we mean by it, since the question touches the deepest points concerning the nature of science. Like most terms referring to mental actions, the verbs *to explain*, or *to explicate*, involve material similes. The action is *ex plicis plana reddere*, to take out the folds, and render a thing plain or even. Explanation thus renders a thing clearly comprehensible in all its points, so that there is nothing left outstanding or obscure.

Every act of explanation consists in pointing out a resemblance between facts, or in showing that similarity exists between apparently diverse phenomena. This similarity may be of any extent and depth; it may be a general law of nature, which harmonises the motions of all the heavenly bodies by showing that there is a similar force which governs all those motions, or the explanation may involve nothing more than a single identity, as when we explain the appearance of shooting stars by showing that they are identical with portions of a comet. Wherever we detect resemblance, there is a more or less explanation. The mind is disquieted when it meets a novel phenomenon, one which is *sui generis*; it seeks at once for parallels which may be found in the memory of past sensations. The so-called sulphurous smell which attends a stroke of lightning often excited attention, and it was not explained until the exact similarity of the smell to that of ozone was pointed out. The marks upon a flagstone are explained when they are shown to correspond with the feet of an extinct animal, whose bones are elsewhere found. Explanation, in fact, generally commences by the discovery of some simple resemblance; the theory of the rainbow began as soon as Antonio de Dominis pointed out the resemblance between its colours and those presented by a ray of sunlight passing through a glass globe full of water.

The nature and limits of explanation can only be fully considered, after we have entered upon the subjects of generalisation and analogy. It must suffice to remark, in this place, that the most important process of explanation consists in showing that an observed fact is one case of a general law or tendency. Iron is always found combined with sulphur, when it is in contact with coal, whereas in other parts of the carboniferous strata it always occurs as a carbonate. We explain this empirical fact as being due to the reducing power of carbon and hydrogen, which prevents the iron from combining with oxygen, and leaves it open to the affinity of sulphur. The uniform strength and direction of the trade-winds were long familiar to mariners, before they were explained by Halley on hydrostatical principles. The winds were found to arise from the action of gravity, which causes a heavier body to displace a lighter one, while the direction from east to west was explained as a result of the earth’s rotation. Whatever body in the northern hemisphere changes its latitude, whether it be a bird, or a railway train, or a body of air, must tend towards the right hand. Dove’s law of the winds is that the winds tend to veer in the northern hemisphere in the direction N.E.S.W., and in the southern hemisphere in the direction N.W.S.E. This tendency was shown by him to be the necessary effect of the same conditions which apply to the trade winds. Whenever, then, any fact is connected by resemblance, law, theory, or hypothesis, with other facts, it is explained.

Although the great mass of recorded facts must be empirical, and awaiting explanation, such knowledge is of minor value, because it does not admit of safe and extensive inference. Each recorded result informs us exactly what will be experienced again in the same circumstances, but has no bearing upon what will happen in other circumstances.

*Overlooked Results of Theory.*

We must by no means suppose that, when a scientific truth is in our possession, all its consequences will be foreseen. Deduction is certain and infallible, in the sense that each step in deductive reasoning will lead us to some result, as certain as the law itself. But it does not follow that deduction will lead the reasoner to every result of a law or combination of laws. Whatever road a traveller takes, he is sure to arrive somewhere, but unless he proceeds in a systematic manner, it is unlikely that he will reach every place to which a network of roads will conduct him.

In like manner there are many phenomena which were virtually within the reach of philosophers by inference from their previous knowledge, but were never discovered until accident or systematic empirical observation disclosed their existence.

That light travels with a uniform high velocity was proved by Roemer from observations of the eclipses of Jupiter’s satellites. Corrections were thenceforward made in all astronomical observations requiring it, for the difference of absolute time at which an event happened, and that at which it would be seen on the earth. But no person happened to remark that the motion of light compounded with that of the earth in its orbit would occasion a small apparent displacement of the greater part of the heavenly bodies. Fifty years elapsed before Bradley empirically discovered this effect, called by him aberration, when reducing his observations of the fixed stars.

When once the relation between an electric current and a magnet had been detected by Oersted and Faraday, it ought to have been possible for them to foresee the diverse results which must ensue in different circumstances. If, for instance, a plate of copper were placed beneath an oscillating magnetic needle, it should have been seen that the needle would induce currents in the copper, but as this could not take place without a certain reaction against the needle, it ought to have been seen that the needle would come to rest more rapidly than in the absence of the copper. This peculiar effect was accidentally discovered by Gambey in 1824. Arago acutely inferred from Gambey’s experiment that if the copper were set in rotation while the needle was stationary the motion would gradually be communicated to the needle. The phenomenon nevertheless puzzled the whole scientific world, and it required the deductive genius of Faraday to show that it was a result of the principles of electro-magnetism.[445]

[445] *Experimental Researches in Electricity*, 1st Series, pp. 24–44.

Many other curious facts might be mentioned which when once noticed were explained as the effects of well-known laws. It was accidentally discovered that the navigation of canals of small depth could be facilitated by increasing the speed of the boats, the resistance being actually reduced by this increase of speed, which enables the boat to ride as it were upon its own forced wave. Now mathematical theory might have predicted this result had the right application of the formulæ occurred to any one.[446] Giffard’s injector for supplying steam boilers with water by the force of their own steam, was, I believe, accidentally discovered, but no new principles of mechanics are involved in it, so that it might have been theoretically invented. The same may be said of the curious experiment in which a stream of air or steam issuing from a pipe is made to hold a free disc upon the end of the pipe and thus obstruct its own outlet. The possession then of a true theory does not by any means imply the foreseeing of all the results. The effects of even a few simple laws may be manifold, and some of the most curious and useful effects may remain undetected until accidental observation brings them to our notice.

[446] Airy, *On Tides and Waves*, Encyclopædia Metropolitana, p. 348*.

*Predicted Discoveries.*

The most interesting of the four classes of facts specified in p. 525, is probably the third, containing those the occurrence of which has been first predicted by theory and then verified by observation. There is no more convincing proof of the soundness of knowledge than that it confers the gift of foresight. Auguste Comte said that “Prevision is the test of true theory;” I should say that it is *one test* of true theory, and that which is most likely to strike the public attention. Coincidence with fact is the test of true theory, but when the result of theory is announced before-hand, there can be no doubt as to the unprejudiced spirit in which the theorist interprets the results of his own theory.

The earliest instance of scientific prophecy is naturally furnished by the science of Astronomy, which was the earliest in development. Herodotus[447] narrates that, in the midst of a battle between the Medes and Lydians, the day was suddenly turned into night, and the event had been foretold by Thales, the Father of Philosophy. A cessation of the combat and peace confirmed by marriages were the consequences of this happy scientific effort. Much controversy has taken place concerning the date of this occurrence, Baily assigning the year 610 B.C., but Airy has calculated that the exact day was the 28th of May, 584 B.C. There can be no doubt that this and other predictions of eclipses attributed to ancient philosophers were due to a knowledge of the Metonic Cycle, a period of 6,585 days, or 223 lunar months, or about 19 years, after which a nearly perfect recurrence of the phases and eclipses of the moon takes place; but if so, Thales must have had access to long series of astronomical records of the Egyptians or the Chaldeans. There is a well-known story as to the happy use which Columbus made of the power of predicting eclipses in overawing the islanders of Jamaica who refused him necessary supplies of food for his fleet. He threatened to deprive them of the moon’s light. “His threat was treated at first with indifference, but when the eclipse actually commenced, the barbarians vied with each other in the production of the necessary supplies for the Spanish fleet.”

[447] Lib. i. cap. 74.

Exactly the same kind of awe which the ancients experienced at the prediction of eclipses, has been felt in modern times concerning the return of comets. Seneca asserted in distinct terms that comets would be found to revolve in periodic orbits and return to sight. The ancient Chaldeans and the Pythagoreans are also said to have entertained a like opinion. But it was not until the age of Newton and Halley that it became possible to calculate the path of a comet in future years. A great comet appeared in 1682, a few years before the first publication of the *Principia*, and Halley showed that its orbit corresponded with that of remarkable comets recorded to have appeared in the years 1531 and 1607. The intervals of time were not quite equal, but Halley conceived the bold idea that this difference might be due to the disturbing power of Jupiter, near which the comet had passed in the interval 1607–1682. He predicted that the comet would return about the end of 1758 or the beginning of 1759, and though Halley did not live to enjoy the sight, it was actually detected on the night of Christmas-day, 1758. A second return of the comet was witnessed in 1835 nearly at the anticipated time.

In recent times the discovery of Neptune has been the most remarkable instance of prevision in astronomical science. A full account of this discovery may be found in several works, as for instance Herschel’s *Outlines of Astronomy*, and *Grant’s History of Physical Astronomy*, Chapters XII and XIII.

*Predictions in the Science of Light.*

Next after astronomy the science of physical optics has furnished the most beautiful instances of the prophetic power of correct theory. These cases are the more striking because they proceed from the profound application of mathematical analysis and show an insight into the mysterious workings of matter which is surprising to all, but especially to those who are unable to comprehend the methods of research employed. By its power of prevision the truth of the undulatory theory of light has been conspicuously proved, and the contrast in this respect between the undulatory and Corpuscular theories is remarkable. Even Newton could get no aid from his corpuscular theory in the invention of new experiments, and to his followers who embraced that theory we owe little or nothing in the science of light. Laplace did not derive from the theory a single discovery. As Fresnel remarks:[448]

[448] Taylor’s *Scientific Memoirs*, vol. v. p. 241.

“The assistance to be derived from a good theory is not to be confined to the calculation of the forces when the laws of the phenomena are known. There are certain laws so complicated and so singular, that observation 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 corpuscular theory had nothing but their own 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, that by stopping a 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, by Billet. Few persons are aware that in the middle 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. 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. 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. An interesting case was observed by Herschel and explained by Airy.[449]

[449] Airy’s *Mathematical Tracts*, 3rd edit. p. 312.

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. As he attributed the double refracting power of crystals to 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 remarkable properties of Iceland spar shown to belong to all transparent substances under certain conditions.[450]

[450] Young’s *Works*, vol. i. p. 412.

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 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 different, but equally strange, effect is produced, a ray of light being spread out into a hollow cone at the point where it quits the crystal. These phenomena are peculiarly interesting, because cones and cylinders of light are not produced in any other cases. They are opposed to all analogy, and constitute singular exceptions, of a kind which we shall afterwards consider more fully. Their 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 wave theory which we owe to Huyghens, Young, and Fresnel.[451]

[451] Lloyd’s *Wave Theory*, Part ii. pp. 52–58. Babbage, *Ninth Bridgewater Treatise*, p. 104, quoting Lloyd, *Transactions of the Royal Irish Academy*, vol. xvii. Clifton, *Quarterly Journal of Pure and Applied Mathematics*, January 1860.

*Predictions from the Theory of Undulations.*

It is curious that the undulations of light, although inconceivably rapid and small, admit of more accurate measurement than waves of any other kind. 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 power of prediction. Herschel was perhaps the first to suggest that two sounds might be made to destroy each other by interference.[452] For if one-half of a wave travelling through a tube could be separated, and conducted by a longer passage, so as, on rejoining the other half, to be one-quarter of a vibration behind-hand, the two portions would exactly neutralise each other. This experiment has been performed with success. 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 holding a vibrating fork close to the ear and turning it round.[453]

[452] *Encyclopædia Metropolitana*, art. *Sound*, p. 753.

[453] Tyndall’s *Sound*, pp. 261, 273.

It is a 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 versâ*, when the relative motion is in the opposite direction, the pitch will be more grave. This arises from the less or greater intervals of time elapsing 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 Buys Ballot, on Dutch railways, and of Scott Russell, in England. Whenever 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 with some express trains it would amount to a tone. A corresponding effect is produced in the case of light undulations, when the eye and the luminous body 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 important and unexpected information concerning the relative approach or recession of stars.

Tides are vast waves, and were the earth’s surface entirely covered by an ocean of uniform depth, they would admit of exact theoretical investigation. The irregular form of the 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 Brill on the coast of Holland, and Lowestoft no tides would be found to exist. At that point the two waves would be of the same amount, but in opposite phases, so as to neutralise each other. This prediction was verified by a surveying vessel of the British navy.[454]

[454] Whewell’s *History of the Inductive Sciences*, vol. ii. p. 471. Herschel’s *Physical Geography*, § 77.

*Prediction in other Sciences.*

Generations, or even centuries, may elapse before mankind are in possession of a mathematical theory of the constitution of matter as complete as the theory of gravitation. Nevertheless, mathematical physicists have in recent years acquired a hold of some of the relations of the physical forces, and the proof is found in anticipations of curious phenomena which had never been observed. Professor James Thomson deduced from Carnot’s theory of heat that the application of pressure would lower the melting-point of ice. He even ventured to assign the amount of this effect, and his statement was afterwards verified by Sir W. Thomson.[455] “In this very remarkable speculation, an entirely novel physical phenomenon was *predicted*, in anticipation of any direct experiments on the subject; and the actual observation of the phenomenon was pointed out as a highly interesting object for experimental research.” Just as liquids which expand in solidifying will have the temperature of solidification lowered by pressure, so liquids which contract in solidifying will exhibit the reverse effect. They will be assisted in solidifying, as it were, by pressure, so as to become solid at a higher temperature, as the pressure is greater. This latter result was verified by Bunsen and Hopkins, in the case of paraffin, spermaceti, wax, and stearin. The effect upon water has more recently been carried to such an extent by Mousson, that under the vast pressure of 1300 atmospheres, water did not freeze until cooled down to -18°C. Another remarkable prediction of Professor Thomson was to the effect that, if a metallic spring be weakened by a rise of temperature, work done against the spring in bending it will cause a cooling effect. Although the effect to be expected in a certain apparatus was only about four-thousandths of a degree Centigrade, Dr. Joule[456] succeeded in measuring it to the extent of three-thousandths of a degree, such is the delicacy of modern heat measurements. I cannot refrain from quoting Dr. Joule’s reflections upon this fact. “Thus even in the above delicate case,” he says, “is the formula of Professor Thomson completely verified. The mathematical investigation of the thermo-elastic qualities of metals has enabled my illustrious friend to predict with certainty a whole class of highly interesting phenomena. To him especially do we owe the important advance which has been recently made to a new era in the history of science, when the famous philosophical system of Bacon will be to a great extent superseded, and when, instead of arriving at discovery by induction from experiment, we shall obtain our largest accessions of new facts by reasoning deductively from fundamental principles.”

[455] Maxwell’s *Theory of Heat*, p. 174. *Philosophical Magazine*, August 1850. Third Series, vol. xxxvii. p. 123.

[456] *Philosophical Transactions*, 1858, vol. cxlviii. p. 127.

The theory of electricity is a necessary part of the general theory of matter, and is rapidly acquiring the power of prevision. As soon as Wheatstone had proved experimentally that the conduction of electricity occupies time, Faraday remarked in 1838, with wonderful sagacity, that if the conducting wires were connected with the coatings of a large Leyden jar, the rapidity of conduction would be lessened. This prediction remained unverified for sixteen years, until the submarine cable was laid beneath the Channel. A considerable retardation of the electric spark was then detected, and Faraday at once pointed out that the wire surrounded by water resembles a Leyden jar on a large scale, so that each message sent through the cable verified his remark of 1838.[457]

[457] Tyndall’s *Faraday*, pp. 73, 74; *Life of Faraday*, vol. ii. pp. 82, 83.

The joint relations of heat and electricity to the metals constitute a new science of thermo-electricity by which Sir W. Thomson was enabled to anticipate the following curious effect, namely, that an electric current passing in an iron bar from a hot to a cold part produces a cooling effect, but in a copper bar the effect is exactly opposite in character, that is, the bar becomes heated.[458] The action of crystals with regard to heat and electricity was partly foreseen on the grounds of theory by Poisson.

[458] Tait’s *Thermodynamics*, p. 77.

Chemistry, although to a great extent an empirical science, has not been without prophetic triumphs. The existence of the metals potassium and sodium was foreseen by Lavoisier, and their elimination by Davy was one of the chief *experimenta crucis* which established Lavoisier’s system. The existence of many other metals which eye had never seen was a natural inference, and theory has not been at fault. In the above cases the compounds of the metal were well known, and it was the result of decomposition that was foretold. The discovery in 1876 of the metal gallium is peculiarly interesting because the existence of this metal, previously wholly unknown, had been inferred from theoretical considerations by M. Mendelief, and some of its properties had been correctly predicted. No sooner, too, had a theory of organic compounds been conceived by Professor A. W. Williamson than he foretold the formation of a complex substance consisting of water in which both atoms of hydrogen are replaced by atoms of acetyle. This substance, known as the acetic anhydride, was afterwards produced by Gerhardt. In the subsequent progress of organic chemistry occurrences of this kind have become common. The theoretical chemist by the classification of his specimens and the manipulation of his formulæ can plan out whole series of unknown oils, acids, and alcohols, just as a designer might draw out a multitude of patterns. Professor Cayley has even calculated for certain cases the possible numbers of chemical compounds.[459] The formation of many such substances is a matter of course; but there is an interesting prediction given by Hofmann, concerning the possible existence of new compounds of sulphur and selenium, and even oxides of ammonium, which it remains for chemists to verify.[460]

[459] *On the Analytical Forms called Trees, with Application to the Theory of Chemical Combinations.* Report of the British Association, 1875, p. 257.

[460] Hofmann’s *Introduction to Chemistry*, pp. 224, 225.

*Prediction by Inversion of Cause and Effect.*

There is one process of experiment which has so often led to important discoveries as to deserve separate illustration--I mean the inversion of Cause and Effect. Thus if A and B in one experiment produce C as a consequent, then antecedents of the nature of B and C may usually be made to produce a consequent of the nature of A inverted in direction. When we apply heat to a gas it tends to expand; hence if we allow the gas to expand by its own elastic force, cold is the result; that is, B (air) and C (expansion) produce the negative of A (heat). Again, B (air) and compression, the negative of C, produce A (heat). Similar results may be expected in a multitude of cases. It is a familiar law that heat expands iron. What may be expected, then, if instead of increasing the length of an iron bar by heat we use mechanical force and stretch the bar? Having the bar and the former consequent, expansion, we should expect the negative of the former antecedent, namely cold. The truth of this inference was proved by Dr. Joule, who investigated the amount of the effect with his usual skill.[461]

[461] *Philosophical Transactions* (1855), vol. cxlv. pp. 100, &c.

This inversion of cause and effect in the case of heat may be itself inverted in a highly curious manner. It happens that there are a few substances which are unexplained exceptions to the general law of expansion by heat. India-rubber especially is remarkable for *contracting* when heated. Since, then, iron and india-rubber are oppositely related to heat, we may expect that as distension of the iron produced cold, distension of the india-rubber will produce heat. This is actually found to be the case, and anyone may detect the effect by suddenly stretching an india-rubber band while the middle part is in the mouth. When being stretched it grows slightly warm, and when relaxed cold.

The reader will see that some of the scientific predictions mentioned in preceding sections were due to the principle of inversion; for instance, Thomson’s speculations on the relation between pressure and the melting-point. But many other illustrations could be adduced. The usual agent by which we melt a substance is heat; but if we can melt a substance without heat, then we may expect the negative of heat as an effect. This is the foundation of all freezing mixtures. The affinity of salt for water causes it to melt ice, and we may thus reduce the temperature to Fahrenheit’s zero. Calcium chloride has so much higher an attraction for water that a temperature of -45° C. may be attained by its use. Even the solution of a certain alloy of lead, tin, and bismuth in mercury, may be made to reduce the temperature through 27° C. All the other modes of producing cold are inversions of more familiar uses of heat. Carré’s freezing machine is an inverted distilling apparatus, the distillation being occasioned by chemical affinity instead of heat. Another kind of freezing machine is the exact inverse of the steam-engine.

A very paradoxical effect is due to another inversion. It is hard to believe that a current of steam at 100° C. can raise a body of liquid to a higher temperature than the steam itself possesses. But Mr. Spence has pointed out that if the boiling-point of a saline solution be above 100°, it will continue, on account of its affinity for water, to condense steam when above 100° in temperature. It will condense the steam until heated to the point at which the tension of its vapour is equal to that of the atmosphere, that is, its own boiling-point.[462] Again, since heat melts ice, we might expect to produce heat by the inverse change from water into ice. This is accomplished in the phenomenon of suspended freezing. Water may be cooled in a clean glass vessel many degrees below the freezing-point, and yet retained in the liquid condition. But if disturbed, and especially if brought into contact with a small particle of ice, it instantly solidifies and rises in temperature to 0° C. The effect is still better displayed in the lecture-room experiment of the suspended crystallisation of a solution of sodium sulphate, in which a sudden rise of temperature of 15° or 20° C. is often manifested.

[462] *Proceedings of the Manchester Philosophical Society*, Feb. 1870.

The science of electricity is full of most interesting cases of inversion. As Professor Tyndall has remarked, Faraday had a profound belief in the reciprocal relations of the physical forces. The great starting-point of his researches, the discovery of electro-magnetism, was clearly an inversion. Oersted and Ampère had proved that with an electric current and a magnet in a particular position as antecedents, motion is the consequent. If then a magnet, a wire and motion be the antecedents, an *opposite* electric current will be the consequent. It would be an endless task to trace out the results of this fertile relationship. Another part of Faraday’s researches was occupied in ascertaining the direct and inverse relations of magnetic and diamagnetic, amorphous and crystalline substances in various circumstances. In all other relations of electricity the principle of inversion holds. The voltameter or the electro-plating cell is the inverse of the galvanic battery. As heat applied to a junction of antimony and bismuth bars produces electricity, it follows that an electric current passed through such a junction will produce cold. But it is now sufficiently apparent that inversion of cause and effect is a most fertile means of discovery and prediction.

*Facts known only by Theory.*

Of the four classes of facts enumerated in p. 525 the last remains unconsidered. It includes the unverified predictions of science. Scientific prophecy arrests the attention of the world when it refers to such striking events as an eclipse, the appearance of a great comet, or any phenomenon which people can verify with their own eyes. But it is surely a matter for greater wonder that a physicist describes and measures phenomena which eye cannot see, nor sense of any kind detect. In most cases this arises from the effect being too small in amount to affect our organs of sense, or come within the powers of our instruments as at present constructed. But there is a class of yet more remarkable cases, in which a phenomenon cannot possibly be observed, and yet we can say what it would be if it were observed.

In astronomy, systematic aberration is an effect of the sun’s proper motion almost certainly known to exist, but which we have no hope of detecting by observation in the present age of the world. As the earth’s motion round the sun combined with the motion of light causes the stars to deviate apparently from their true positions to the extent of about 18″ at the most, so the motion of the whole planetary system through space must occasion a similar displacement of at most 5″. The ordinary aberration can be readily detected with modern astronomical instruments, because it goes through a yearly change in direction or amount; but systematic aberration is constant so long as the planetary system moves uniformly in a sensibly straight line. Only then in the course of ages, when the curvature of the sun’s path becomes apparent, can we hope to verify the existence of this kind of aberration. A curious effect must also be produced by the sun’s proper motion upon the apparent periods of revolution of the binary stars.

To my mind, some of the most interesting truths in the whole range of science are those which have not been, and in many cases probably never can be, verified by trial. Thus the chemist assigns, with a very high degree of probability, the vapour densities of such elements as carbon and silicon, which have never been observed separately in a state of vapour. The chemist is also familiar with the vapour densities of elements at temperatures at which the elements in question never have been, and probably never can be, submitted to experiment in the form of vapour.

Joule and others have calculated the actual velocity of the molecules of a gas, and even the number of collisions which must take place per second during their constant circulation. Physicists have not yet given us the exact magnitudes of the particles of matter, but they have ascertained by several methods the limits within which their magnitudes must lie. Such scientific results must be for ever beyond the power of verification by the senses. I have elsewhere had occasion to remark that waves of light, the intimate processes of electrical changes, the properties of the ether which is the base of all phenomena, are necessarily determined in a hypothetical, but not therefore a less certain manner.

Though only two of the metals, gold and silver, have ever been observed to be transparent, we know on the grounds of theory that they are all more or less so; we can even estimate by theory their refractive indices, and prove that they are exceedingly high. The phenomena of elliptic polarisation, and perhaps also those of internal radiation,[463] depend upon the refractive index, and thus, even when we cannot observe any refracted rays, we can indirectly learn how they would be refracted.

[463] Balfour Stewart, *Elementary Treatise on Heat*, 1st edit. p. 198.

In many cases large quantities of electricity must be produced, which we cannot observe because it is instantly discharged. In the common electric machine the cylinder and rubber are made of non-conductors, so that we can separate and accumulate the electricity. But a little damp, by serving as a conductor, prevents this separation from enduring any sensible time. Hence there is no doubt that when we rub two good conductors against each other, for instance two pieces of metals, much electricity is produced, but instantaneously converted into some other form of energy. Joule believes that all the heat of friction is transmuted electricity.

As regards phenomena of insensible amount, nature is absolutely full of them. We must regard those changes which we can observe as the comparatively rare aggregates of minuter changes. On a little reflection we must allow that no object known to us remains for two instants of exactly the same temperature. If so, the dimensions of objects must be in a perpetual state of variation. The minor planetary and lunar perturbations are infinitely numerous, but usually too small to be detected by observation, although their amounts may be assigned by theory. There is every reason to believe that chemical and electric actions of small amount are constantly in progress. The hardest substances, if reduced to extremely small particles, and diffused in pure water, manifest oscillatory movements which must be due to chemical and electric changes, so slight that they go on for years without affecting appreciably the weight of the particles.[464] The earth’s magnetism must more or less affect every object which we handle. As Tyndall remarks, “An upright iron stone influenced by the earth’s magnetism becomes a magnet, with its bottom a north and its top a south pole. Doubtless, though in an immensely feebler degree, every erect marble statue is a true diamagnet, with its head a north pole and its feet a south pole. The same is certainly true of man as he stands upon the earth’s surface, for all the tissues of the human body are diamagnetic.”[465] The sun’s light produces a very quick and perceptible effect upon the photographic plate; in all probability it has a less effect upon a great variety of substances. We may regard every phenomenon as an exaggerated and conspicuous case of a process which is, in infinitely numerous cases, beyond the means of observation.

[464] Jevons, *Proceedings of the Manchester Literary and Philosophical Society*, 25th January, 1870, vol. ix. p. 78.

[465] *Philosophical Transactions*, vol. cxlvi. p. 249.