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.Return to passage
[465] *Philosophical Transactions*, vol. cxlvi. p. 249.Return to passage
XXV. ACCORDANCE OF QUANTITATIVE THEORIES
In the preceding chapter we found that facts may be classed under four heads as regards their connection with theory, and our powers of explanation or prediction. The facts hitherto considered were generally of a qualitative rather than a quantitative nature; but when we look exclusively to the quantity of a phenomenon, and the various modes in which we may determine its amount, nearly the same system of classification will hold good. There will, however, be five possible cases:--
(1) We may directly and empirically measure a phenomenon, without being able to explain why it should have any particular quantity, or to connect it by theory with other quantities.
(2) In a considerable number of cases we can theoretically predict the existence of a phenomenon, but are unable to assign its amount, except by direct measurement, or to explain the amount theoretically when thus ascertained.
(3) We may measure a quantity, and afterwards explain it as related to other quantities, or as governed by known quantitative laws.
(4) We may predict the quantity of an effect on theoretical grounds, and afterwards confirm the prediction by direct measurement.
(5) We may indirectly determine the quantity of an effect without being able to verity it by experiment.
These classes of quantitative facts might be illustrated by an immense number of interesting points in the history of physical science. Only a few instances of each class can be given here.
*Empirical Measurements.*
Under the first head of purely empirical measurements, which have not been brought under any theoretical system, may be placed the great bulk of quantitative facts recorded by scientific observers. The tables of numerical results which abound in books on chemistry and physics, the huge quartos containing the observations of public observatories, the multitudinous tables of meteorological observations, which are continually being published, the more abstruse results concerning terrestrial magnetism--such results of measurement, for the most part, remain empirical, either because theory is defective, or the labour of calculation and comparison is too formidable. In the Greenwich Observatory, indeed, the salutary practice has been maintained by the present Astronomer Royal, of always reducing the observations, and comparing them with the theories of the several bodies. The divergences from theory thus afford material for the discovery of errors or of new phenomena; in short, the observations have been turned to the use for which they were intended. But it is to be feared that other establishments are too often engaged in merely recording numbers of which no real use is made, because the labour of reduction and comparison with theory is too great for private inquirers to undertake. In meteorology, especially, great waste of labour and money is taking place, only a small fraction of the results recorded being ever used for the advancement of the science. For one meteorologist like Quetelet, Dove, or Baxendell, who devotes himself to the truly useful labour of reducing other people’s observations, there are hundreds who labour under the delusion that they are advancing science by loading our book-shelves with numerical tables. It is to be feared, in like manner, that almost the whole bulk of statistical numbers, whether commercial, vital, or moral, is of little scientific value. Purely empirical measurements may have a direct practical value, as when tables of the specific gravity, or strength of materials, assist the engineer; the specific gravities of mixtures of water with acids, alcohols, salts, &c., are useful in chemical manufactories, custom-house gauging, &c.; observations of rainfall are requisite for questions of water supply; the refractive index of various kinds of glass must be known in making achromatic lenses; but in all such cases the use made of the measurements is not scientific but practical. It may be asserted, that no number which remains isolated, and uncompared by theory with other numbers, is of scientific value. Having tried the tensile strength of a piece of iron in a particular condition, we know what will be the strength of the same kind of iron in a similar condition, provided we can ever meet with that exact kind of iron again; but we cannot argue from piece to piece, nor lay down any laws exactly connecting the strength of iron with the quantity of its impurities.
*Quantities indicated by Theory, but Empirically Measured.*
In many cases we are able to foresee the existence of a quantitative effect, on the ground of general principles, but are unable, either from the want of numerical data, or from the entire absence of any mathematical theory, to assign the amount of such effect. We then have recourse to direct experiment to determine its amount. Whether we argued from the oceanic tides by analogy, or deductively from the theory of gravitation, there could be no doubt that atmospheric tides of some amount must occur in the atmosphere. Theory, however, even in the hands of Laplace, was not able to overcome the complicated mechanical conditions of the atmosphere, and predict the amounts of such tides; and, on the other hand, these amounts were so small, and were so masked by far larger undulations arising from the heating power of the sun, and from other meteorological disturbances, that they would probably have never been discovered by purely empirical observations. Theory having, however, indicated their existence and their periods, it was easy to make series of barometrical observations in places selected so as to be as free as possible from casual fluctuations, and then, by the suitable application of the method of means, to detect the small effects in question. The principal lunar atmospheric tide was thus proved to amount to between ·003 and ·004 inch.[466]
[466] Grant’s *History of Physical Astronomy*, p. 162.
Theory yields the greatest possible assistance in applying the method of means. For if we have a great number of empirical measurements, each representing the joint effect of a number of causes, our object will be to take the mean of all those in which the effect to be measured is present, and compare it with the mean of the remainder in which the effect is absent, or acts in the opposite direction. The difference will then represent the amount of the effect, or double the amount respectively. Thus, in the case of the atmospheric tides, we take the mean of all the observations when the moon was on the meridian, and compare it with the mean of all observations when she was on the horizon. In this case we trust to chance that all other effects will lie about as often in one direction as the other, and will neutralise themselves in the drawing of each mean. It is a great advantage, however, to be able to decide by theory when each principal disturbing effect is present or absent; for the means may then be drawn so as to separate each such effect, leaving only minor and casual divergences to the law of error. Thus, if there be three principal effects, and we draw means giving respectively the sum of all three, the sum of the first two, and the sum of the last two, then we gain three simple equations, by the solution of which each quantity is determined.
*Explained Results of Measurement.*
The second class of measured phenomena contains those which, after being determined in a direct and purely empirical application of measuring instruments, are afterwards shown to agree with some hypothetical explanation. Such results are turned to their proper use, and several advantages may arise from the comparison. The correspondence with theory will seldom or never be precise; and, even if it be so, the coincidence must be regarded as accidental.
If the divergences between theory and experiment be comparatively small, and variable in amount and direction, they may often be safely attributed to inconsiderable sources of error in the experimental processes. The strict method of procedure is to calculate the probable error of the mean of the observed results (p. 387), and then observe whether the theoretical result falls within the limits of probable error. If it does, and if the experimental results agree as well with theory as they agree with each other, then the probability of the theory is much increased, and we may employ the theory with more confidence in the anticipation of further results. The probable error, it should be remembered, gives a measure only of the effects of incidental and variable sources of error, but in no degree indicates the amount of fixed causes of error. Thus, if the mean results of two modes of determining a quantity are so far apart that the limits of probable error do not overlap, we may infer the existence of some overlooked source of fixed error in one or both modes. We will further consider in a subsequent section the discordance of measurements.
*Quantities determined by Theory and verified by Measurement.*
One of the most satisfactory tests of a theory consists in its application not only to predict the nature of a phenomenon, and the circumstances in which it may be observed, but also to assign the precise quantity of the phenomenon. If we can subsequently apply accurate instruments and measure the amount of the phenomenon witnessed, we have an excellent opportunity of verifying or negativing the theory. It was in this manner that Newton first attempted to verify his theory of gravitation. He knew approximately the velocity produced in falling bodies at the earth’s surface, and if the law of the inverse square of the distance held true, and the reputed distance of the moon was correct, he could infer that the moon ought to fall towards the earth at the rate of fifteen feet in one minute. Now, the actual divergence of the moon from the tangent of its orbit appeared to amount only to thirteen feet in one minute, and there was a discrepancy of two feet in fifteen, which caused Newton to lay “aside at that time any further thoughts of this matter.” Many years afterwards, probably fifteen or sixteen years, Newton obtained more precise data from which he could calculate the size of the moon’s orbit, and he then found the discrepancy to be inconsiderable.
His theory of gravitation was thus verified as far as the moon was concerned; but this was to him only the beginning of a long course of deductive calculations, each ending in a verification. If the earth and moon attract each other, and also the sun and the earth, there is reason to expect that the sun and moon should attract each other. Newton followed out the consequences of this inference, and showed that the moon would not move as if attracted by the earth only, but sometimes faster and sometimes slower. Comparison with Flamsteed’s observations of the moon showed that such was the case. Newton argued again, that as the waters of the ocean are not rigidly attached to the earth, they might attract the moon, and be attracted in return, independently of the rest of the earth. Certain daily motions resembling the tides would then be caused, and there were the tides to verify the reasoning. It was the extraordinary power with which Newton traced out geometrically the consequences of his theory, and submitted them to repeated comparison with experience, which constitutes his pre-eminence over all physicists.
*Quantities determined by Theory and not verified.*
It will continually happen that we are able, from certain measured phenomena and a correct theory, to determine the amount of some other phenomenon which we may either be unable to measure at all, or to measure with an accuracy corresponding to that required to verify the prediction. Thus Laplace having worked out a theory of the motions of Jupiter’s satellites on the hypothesis of gravitation, found that these motions were greatly affected by the spheroidal form of Jupiter. The motions of the satellites can be observed with great accuracy owing to their frequent eclipses and transits, and from these motions he was able to argue inversely, and assign the ellipticity of the planet. The ratio of the polar and equatorial axes thus determined was very nearly that of 13 to 14; and it agrees well with such direct micrometrical measurements of the planet as have been made; but Laplace believed that the theory gave a more accurate result than direct observation could yield, so that the theory could hardly be said to admit of direct verification.
The specific heat of air was believed on the grounds of direct experiment to amount to 0·2669, the specific heat of water being taken as unity; but the methods of experiment were open to considerable causes of error. Rankine showed in 1850 that it was possible to calculate from the mechanical equivalent of heat and other thermodynamic data, what this number should be, and he found it to be 0·2378. This determination was at the time accepted as the most satisfactory result, although not verified; subsequently in 1853 Regnault obtained by direct experiment the number 0·2377, proving that the prediction had been well grounded.
It is readily seen that in quantitative questions verification is a matter of degree and probability. A less accurate method of measurement cannot verify the results of a more accurate method, so that if we arrive at a determination of the same physical quantity in several distinct modes it is often a delicate matter to decide which result is most reliable, and should be used for the indirect determination of other quantities. For instance, Joule’s and Thomson’s ingenious experiments upon the thermal phenomena of fluids in motion[467] involved, as one physical constant, the mechanical equivalent of heat; if requisite, then, they might have been used to determine that important constant. But if more direct methods of experiment give the mechanical equivalent of heat with superior accuracy, then the experiments on fluids will be turned to a better use in determining various quantities relating to the theory of fluids. We will further consider questions of this kind in succeeding sections.
[467] *Philosophical Transactions* (1854), vol. cxliv. p. 364.
There are of course many quantities assigned on theoretical grounds which we are quite unable to verify with corresponding accuracy. The thickness of a film of gold leaf, the average depths of the oceans, the velocity of a star’s approach to or regression from the earth as inferred from spectroscopic data (pp. 296–99), are cases in point; but many others might be quoted where direct verification seems impossible. Newton and subsequent physicists have measured light undulations, and by several methods we learn the velocity with which light travels. Since an undulation of the middle green is about five ten-millionths of a metre in length, and travels at the rate of nearly 300,000,000 of metres per second, it follows that about 600,000,000,000,000 undulations must strike in one second the retina of an eye which perceives such light. But how are we to verify such an astounding calculation by directly counting pulses which recur six hundred billions of times in a second?
*Discordance of Theory and Experiment.*
When a distinct want of accordance is found to exist between the results of theory and direct measurement, interesting questions arise as to the mode in which we can account for this discordance. The ultimate explanation of the discrepancy may be accomplished in at least four ways as follows:--
(1) The direct measurement may be erroneous owing to various sources of casual error.
(2) The theory may be correct as far as regards the general form of the supposed laws, but some of the constant numbers or other quantitative data employed in the theoretical calculations may be inaccurate.
(3) The theory may be false, in the sense that the forms of the mathematical equations assumed to express the laws of nature are incorrect.
(4) The theory and the involved quantities may be approximately accurate, but some regular unknown cause may have interfered, so that the divergence may be regarded as a *residual effect* representing possibly a new and interesting phenomenon.
No precise rules can be laid down as to the best mode of proceeding to explain the divergence, and the experimentalist will have to depend upon his own insight and knowledge; but the following recommendations may be made.
If the experimental measurements are not numerous, repeat them and take a more extensive mean result, the probable accuracy of which, as regards casual errors, will increase as the square root of the number of experiments. Supposing that no considerable modification of the result is thus effected, we may suspect the existence of more deep-seated sources of error in our method of measurement. The next resource will be to change the size and form of the apparatus employed, and to introduce various modifications in the materials employed or the course of procedure, in the hope (p. 396) that some cause of constant error may thus be removed. If the inconsistency with theory still remains unreduced we may attempt to invent some widely different mode of arriving at the same physical quantity, so that we may be almost sure that the same cause of error will not affect both the new and old results. In some cases it is possible to find five or six essentially different modes of arriving at the same determination.
Supposing that the discrepancy still exists we may begin to suspect that our direct measurements are correct, and that the data employed in the theoretical calculations are inaccurate. We must now review the grounds on which these data depend, consisting as they must ultimately do of direct measurements. A comparison of the recorded data will show the degree of probability attaching to the mean result employed; and if there is any ground for imagining the existence of error, we should repeat the observations, and vary the forms of experiment just as in the case of the previous direct measurements. The continued existence of the discrepancy must show that we have not attained to a complete acquaintance with the theory of the causes in action, but two different cases still remain. We may have misunderstood the action of those causes which we know to exist, or we may have overlooked the existence of one or more other causes. In the first case our hypothesis appears to be wrongly chosen and inapplicable; but whether we are to reject it will depend upon whether we can form another hypothesis which yields a more accurate accordance. The probability of an hypothesis, it will be remembered (p. 243), is to be judged, in the absence of *à priori* grounds of judgment, by the probability that if the supposed causes exist the observed result follows; but as there is now little probability of reconciling the original hypothesis with our direct measurements the field is open for new hypotheses, and any one which gives a closer accordance with measurement will so far have better claims to attention. Of course we must never estimate the probability of an hypothesis merely by its accordance with a few results only. Its general analogy and accordance with other known laws of nature, and the fact that it does not conflict with other probable theories, must be taken into account, as we shall see in the next book. The requisite condition of a good hypothesis, that it must admit of the deduction of facts verified in observation, must be interpreted in the widest manner, as including all ways in which there may be accordance or discordance. All our attempts at reconciliation having failed, the only conclusion we can come to is that some unknown cause of a new character exists. If the measurements be accurate and the theory probable, then there remains a *residual phenomenon*, which, being devoid of theoretical explanation, must be set down as a new empirical fact worthy of further investigation. Outstanding residual discrepancies have often been found to involve new discoveries of the greatest importance.
*Accordance of Measurements of Astronomical Distances.*
One of the most instructive instances which we can meet, of the manner in which different measurements confirm or check each other, is furnished by the determination of the velocity of light, and the dimensions of the planetary system. Roemer first discovered that light requires time to travel, by observing that the eclipses of Jupiter’s satellites, although they occur at fixed moments of absolute time, are visible at different moments in different parts of the earth’s orbit, according to the distance between the earth and Jupiter. The time occupied by light in traversing the mean semi-diameter of the earth’s orbit is found to be about eight minutes. The mean distance of the sun and earth was long assumed by astronomers as being about 95,274,000 miles, this result being deduced by Bessel from the observations of the transit of Venus, which occurred in 1769, and which were found to give the solar parallax, or which is the same thing, the apparent angular magnitude of the earth seen from the sun, as equal to 8″·578. Dividing the mean distance of the sun and earth by the number of seconds in 8^{m}. 13^{s}.3 we find the velocity of light to be about 192,000 miles per second.
Nearly the same result was obtained in what seems a different manner. The aberration of light is the apparent change in the direction of a ray of light owing to the composition of its motion with that of the earth’s motion round the sun. If we know the amount of aberration and the mean velocity of the earth, we can estimate that of light, which is thus found to be 191,100 miles per second. Now this determination depends upon a new physical quantity, that of aberration, which is ascertained by direct observation of the stars, so that the close accordance of the estimates of the velocity of light as thus arrived at by different methods might seem to leave little room for doubt, the difference being less than one per cent.
Nevertheless, experimentalists were not satisfied until they had succeeded in measuring the velocity of light by direct experiments performed upon the earth’s surface. Fizeau, by a rapidly revolving toothed wheel, estimated the velocity at 195,920 miles per second. As this result differed by about one part in sixty from estimates previously accepted, there was thought to be room for further investigation. The revolving mirror, used by Wheatstone in measuring the velocity of electricity, was now applied in a more refined manner by Fizeau and by Foucault to determine the velocity of light. The latter physicist came to the startling conclusion that the velocity was not really more than 185,172 miles per second. No repetition of the experiment would shake this result, and there was accordingly a discrepancy between the astronomical and the experimental results of about 7,000 miles per second. The latest experiments, those of M. Cornu, only slightly raise the estimate, giving 186,660 miles per second. A little consideration shows that both the astronomical determinations involve the magnitude of the earth’s orbit as one datum, because our estimate of the earth’s velocity in its orbit depends upon our estimate of the sun’s mean distance. Accordingly as regards this quantity the two astronomical results count only for one. Though the transit of Venus had been considered to give the best data for the calculation of the sun’s parallax, yet astronomers had not neglected less favourable opportunities. Hansen, calculating from certain inequalities in the moon’s motion, had estimated it at 8″·916; Winneke, from observations of Mars, at 8″·964; Leverrier, from the motions of Mars, Venus, and the moon, at 8″·950. These independent results agree much better with each other than with that of Bessel (8″·578) previously received, or that of Encke (8″·58) deduced from the transits of Venus in 1761 and 1769, and though each separately might be worthy of less credit, yet their close accordance renders their mean result (8″·943) comparable in probability with that of Bessel. It was further found that if Foucault’s value for the velocity of light were assumed to be correct, and the sun’s distance were inversely calculated from that, the sun’s parallax would be 8″·960, which closely agreed with the above mean result. This further correspondence of independent results threw the balance of probability strongly against the results of the transit of Venus, and rendered it desirable to reconsider the observations made on that occasion. Mr. E. J. Stone, having re-discussed those observations,[468] found that grave oversights had been made in the calculations, which being corrected would alter the estimate of parallax to 8″·91, a quantity in such comparatively close accordance with the other results that astronomers did not hesitate at once to reduce their estimate of the sun’s mean distance from 95,274,000 to 91,771,000, miles, although this alteration involved a corresponding correction in the assumed magnitudes and distances of most of the heavenly bodies. The solar parallax is now (1875) believed to be about 8″·878, the number deduced from Cornu’s experiments on the velocity of light. This result agrees very closely with 8″·879, the estimate obtained from new observations on the transit of Venus, by the French observers, and with 8″·873, the result of Galle’s observations of the planet Flora. When all the observations of the late transit of Venus are fully discussed the sun’s distance will probably be known to less than one part in a thousand, if not one part in ten thousand.[469]
[468] *Monthly Notices of the Royal Astronomical Society*, vol. xxviii. p. 264.
[469] It would seem to be absurd to repeat the profuse expenditure of 1874 at the approaching transit in 1882. The aggregate sum spent in 1874 by various governments and individuals can hardly be less than £200,000, a sum which, wisely expended on scientific investigations, would give a hundred important results.
In this question the theoretical relations between the velocity of light, the constant of aberration, the sun’s parallax, and the sun’s mean distance, are of the simplest character, and can hardly be open to any doubt, so that the only doubt was as to which result of observation was the most reliable. Eventually the chief discrepancy was found to arise from misapprehension in the reduction of observations, but we have a satisfactory example of the value of different methods of estimation in leading to the detection of a serious error. Is it not surprising that Foucault by measuring the velocity of light when passing through the space of a few yards, should lead the way to a change in our estimates of the magnitudes of the whole universe?
*Selection of the best Mode of Measurement.*
When we once obtain command over a question of physical science by comprehending the theory of the subject, we often have a wide choice opened to us as regards the methods of measurement, which may thenceforth be made to give the most accurate results. If we can measure one fundamental quantity very precisely we may be able by theory to determine accurately many other quantitative results. Thus, if we determine satisfactorily the atomic weights of certain elements, we do not need to determine with equal accuracy the composition and atomic weights of their several compounds. Having learnt the relative atomic weights of oxygen and sulphur, we can calculate the composition by weight of the several oxides of sulphur. Chemists accordingly select with the greatest care that compound of two elements which seems to allow of the most accurate analysis, so as to give the ratio of their atomic weights. It is obvious that we only need the ratio of the atomic weight of each element to that of some common element, in order to calculate, that of each to each. Moreover the atomic weight stands in simple relation to other quantitative facts. The weights of equal volumes of elementary gases at equal temperature and pressure have the same ratios as the atomic weights; now, as nitrogen under such circumstances weighs 14·06 times as much as hydrogen, we may infer that the atomic weight of nitrogen is about 14·06, or more probably 14·00, that of hydrogen being unity. There is much evidence, again, that the specific heats of elements are inversely as their atomic weights, so that these two classes of quantitative data throw light mutually upon each other. In fact the atomic weight, the atomic volume, and the atomic heat of an element, are quantities so closely connected that the determination of one will lead to that of the others. The chemist has to solve a complicated problem in deciding in the case of each of 60 or 70 elements which mode of determination is most accurate. Modern chemistry presents us with an almost infinitely extensive web of numerical ratios developed out of a few fundamental ratios.
In hygrometry we have a choice among at least four modes of measuring the quantity of aqueous vapour contained in a given bulk of air. We can extract the vapour by absorption in sulphuric acid, and directly weigh its amount; we can place the air in a barometer tube and observe how much the absorption of the vapour alters the elastic force of the air; we can observe the dew-point of the air, that is the temperature at which the vapour becomes saturated; or, lastly, we can insert a dry and wet bulb thermometer and observe the temperature of an evaporating surface. The results of each mode can be connected by theory with those of the other modes, and we can select for each experiment that mode which is most accurate or most convenient. The chemical method of direct measurement is capable of the greatest accuracy, but is troublesome; the dry and wet bulb thermometer is sufficiently exact for meteorological purposes and is most easy to use.
*Agreement of Distinct Modes of Measurement.*
Many illustrations might be given of the accordance which has been found to exist in some cases between the results of entirely different methods of arriving at the measurement of a physical quantity. While such accordance must, in the absence of information to the contrary, be regarded as the best possible proof of the approximate correctness of the mean result, yet instances have occurred to show that we can never take too much trouble in confirming results of great importance. When three or even more distinct methods have given nearly coincident numbers, a new method has sometimes disclosed a discrepancy which it is yet impossible to explain.
The ellipticity of the earth is known with considerable approach to certainty and accuracy, for it has been estimated in three independent ways. The most direct mode is to measure long arcs extending north and south upon the earth’s surface, by means of trigonometrical surveys, and then to compare the lengths of these arcs with their curvature as determined by observations of the altitude of certain stars at the terminal points. The most probable ellipticity of the earth deduced from all measurements of this kind was estimated by Bessel at 1/300, though subsequent measurements might lead to a slightly different estimate. The divergence from a globular form causes a small variation in the force of gravity at different parts of the earth’s surface, so that exact pendulum observations give the data for an independent estimate of the ellipticity, which is thus found to be 1/320. In the third place the spheroidal protuberance about the earth’s equator leads to a certain inequality in the moon’s motion, as shown by Laplace; and from the amount of that inequality, as given by observations, Laplace was enabled to calculate back to the amount of its cause. He thus inferred that the ellipticity is 1/305, which lies between the two numbers previously given, and was considered by him the most satisfactory determination. In this case the accordance is undisturbed by subsequent results, so that we are obliged to accept Laplace’s result as a highly probable one.
The mean density of the earth is a constant of high importance, because it is necessary for the determination of the masses of all the other heavenly bodies. Astronomers and physicists accordingly have bestowed a great deal of labour upon the exact estimation of this constant. The method of procedure consists in comparing the gravitation of the globe with that of some body of matter of which the mass is known in terms of the assumed unit of mass. This body of matter, serving as an intermediate term of comparison, may be variously chosen; it may consist of a mountain, or a portion of the earth’s crust, or a heavy ball of metal. The method of experiment varies so much according as we select one body or the other, that we may be said to have three independent modes of arriving at the desired result.
The mutual gravitation of two balls is so exceedingly small compared with their gravitation towards the immense mass of the earth, that it is usually quite imperceptible, and although asserted by Newton to exist, on the ground of theory, was never observed until the end of the 18th century. Michell attached two small balls to the extremities of a delicately suspended torsion balance, and then bringing heavy balls of lead alternately to either side of these small balls was able to detect a slight deflection of the torsion balance. He thus furnished a new verification of the theory of gravitation. Cavendish carried out the experiment with more care, and estimated the gravitation of the balls by treating the torsion balance as a pendulum; then taking into account the respective distances of the balls from each other and from the centre of the earth, he was able to assign 5·48 (or as re-computed by Baily, 5·448) as the probable mean density of the earth. Newton’s sagacious guess to the effect that the density of the earth was between five and six times that of water, was thus remarkably confirmed. The same kind of experiment repeated by Reich gave 5·438. Baily having again performed the experiment with every possible refinement obtained a slightly higher number, 5·660.
A different method of procedure consisted in ascertaining the effect of a mountain mass in deflecting the plumb-line; for, assuming that we can determine the dimensions and mean density of the mountain, the plumb-line enables us to compare its mass with that of the whole earth. The mountain Schehallien was selected for the experiment, and observations and calculations performed by Maskelyne, Hutton, and Playfair, gave as the most probable result 4·713. The difference from the experimental results already mentioned is considerable and is important, because the instrumental operations are of an entirely different character from those of Cavendish and Baily’s experiments. Sir Henry James’ similar determination from the attraction of Arthur’s Seat gave 5·14.
A third distinct method consists in determining the force of gravity at points elevated above the surface of the earth on mountain ranges, or sunk below it in mines. Carlini experimented with a pendulum at the hospice of Mont Cenis, 6,375 feet above the sea, and by comparing the attractive forces of the earth and the Alps, found the density to be still smaller, namely, 4·39, or as corrected by Giulio, 4·950. Lastly, the Astronomer Royal has on two occasions adopted the opposite method of observing a pendulum at the bottom of a deep mine, so as to compare the density of the strata penetrated with the density of the whole earth. On the second occasion he carried his method into effect at the Harton Colliery, 1,260 feet deep; all that could be done by skill in measurement and careful consideration of all the causes of error, was accomplished in this elaborate series of observations[470] (p. 291). No doubt Sir George Airy was much perplexed when he found that his new result considerably exceeded that obtained by any other method, being no less than 6·566, or 6·623 as finally corrected. In this case we learn an impressive lesson concerning the value of repeated determinations by distinct methods in disabusing our minds of the reliance which we are only too apt to place in results which show a certain degree of coincidence.
[470] *Philosophical Transactions* (1856), vol. cxlvi. p. 342.
In 1844 Herschel remarked in his memoir of Francis Baily,[471] “that the mean specific gravity of this our planet is, in all human probability, quite as well determined as that of an ordinary hand-specimen in a mineralogical cabinet,--a marvellous result, which should teach us to despair of nothing which lies within the compass of number, weight and measure.” But at the same time he pointed out that Baily’s final result, of which the probable error was only 0·0032, was the highest of all determinations then known, and Airy’s investigation has since given a much higher result, quite beyond the limits of probable error of any of the previous experiments. If we treat all determinations yet made as of equal weight, the simple mean is about 5·45, the mean error nearly 0·5, and the probable error almost 0·2, so that it is as likely as not that the truth lies between 5·65 and 5·25 on this view of the matter. But it is remarkable that the two most recent and careful series of observations by Baily and Airy,[472] lie beyond these limits, and as with the increase of care the estimate rises, it seems requisite to reject the earlier results, and look upon the question as still requiring further investigation. Physicists often take 5-2/3 or 5·67 as the best guess at the truth, but it is evident that new experiments are much required. I cannot help thinking that a portion of the great sums of money which many governments and private individuals spent upon the transit of Venus expeditions in 1874, and which they will probably spend again in 1882 (p. 562), would be better appropriated to new determinations of the earth’s density. It seems desirable to repeat Baily’s experiment in a vacuous case, and with the greater mechanical refinements which the progress of the last forty years places at the disposal of the experimentalist. It would be desirable, also, to renew the pendulum experiments of Airy in some other deep mine. It might even be well to repeat upon some suitable mountain the observations performed at Schehallien. All these operations might be carried out for the cost of one of the superfluous transit expeditions.
[471] *Monthly Notices of the Royal Astronomical Society*, for 8th Nov. 1844, No. X. vol. vi. p. 89.
[472] *Philosophical Magazine*, 2nd Series, vol. xxvi. p. 61.
Since the establishment of the dynamical theory of heat it has become a matter of the greatest importance to determine with accuracy the mechanical equivalent of heat, or the quantity of energy which must be given, or received, in a definite change of temperature effected in a definite quantity of a standard substance, such as water. No less than seven almost entirely distinct modes of determining this constant have been tried. Dr. Joule first ascertained by the friction of water that to raise the temperature of one kilogram of water through one degree centigrade, we must employ energy sufficient to raise 424 kilograms through the height of one metre against the force of gravity at the earth’s surface. Joule, Mayer, Clausius,[473] Favre and other experimentalists have made determinations by less direct methods. Experiments on the mechanical properties of gases give 426 kilogrammetres as the constant; the work done by a steam-engine gives 413; from the heat evolved in electrical experiments several determinations have been obtained; thus from induced electric currents we get 452; from the electro-magnetic engine 443; from the circuit of a battery 420; and, from an electric current, the lowest result of all, namely, 400.[474]
[473] Clausius in *Philosophical Magazine*, 4th Series, vol. ii. p. 119.
[474] Watts’ *Dictionary of Chemistry*, vol. iii. p. 129.
Considering the diverse and in many cases difficult methods of observation, these results exhibit satisfactory accordance, and their mean (423·9) comes very close to the number derived by Dr. Joule from the apparently most accurate method. The constant generally assumed as the most probable result is 423·55 kilogrammetres.
*Residual Phenomena.*
Even when the experimental data employed in the verification of a theory are sufficiently accurate, and the theory itself is sound, there may exist discrepancies demanding further investigation. Herschel pointed out the importance of such outstanding quantities, and called them *residual phenomena*.[475] Now if the observations and the theory be really correct, such discrepancies must be due to the incompleteness of our knowledge of the causes in action, and the ultimate explanation must consist in showing that there is in action, either
[475] *Preliminary Discourse*, §§ 158, 174. *Outlines of Astronomy*, 4th edit. § 856.
(1) Some agent of known nature whose presence was not suspected;
Or (2) Some new agent of unknown nature.
In the first case we can hardly be said to make a new discovery, for our ultimate success consists merely in reconciling the theory with known facts when our investigation is more comprehensive. But in the second case we meet with a totally new fact, which may lead us to realms of new discovery. Take the instance adduced by Herschel. The theory of Newton and Halley concerning comets was that they were gravitating bodies revolving round the sun in elliptic orbits, and the return of Halley’s Comet, in 1758, verified this theory. But, when accurate observations of Encke’s Comet came to be made, the verification was not found to be exact. Encke’s Comet returned each time a little sooner than it ought to do, the period regularly decreasing from 1212·79 days, between 1786 and 1789, to 1210·44 between 1855 and 1858; and the hypothesis has been started that there is a resisting medium filling the space through which the comet passes. This hypothesis is a *deus ex machinâ* for explaining this solitary phenomenon, and cannot possess much probability unless it can be shown that other phenomena are deducible from it. Many persons have identified this medium with that through which light undulations pass, but I am not aware that there is anything in the undulatory theory of light to show that the medium would offer resistance to a moving body. If Professor Balfour Stewart can prove that a rotating disc would experience resistance in a vacuous receiver, here is an experimental fact which distinctly supports the hypothesis. But in the mean time it is open to question whether other known agents, for instance electricity, may not be brought in, and I have tried to show that if, as is believed, the tail of a comet is an electrical phenomenon, it is a necessary result of the conservation of energy that the comet shall exhibit a loss of energy manifested in a diminution of its mean distance from the sun and its period of revolution.[476] It should be added that if Professor Tait’s theory be correct, as seems very probable, and comets consist of swarms of small meteors, there is no difficulty in accounting for the retardation. It has long been known that a collection of small bodies travelling together in an orbit round a central body will tend to fall towards it. In either case, then, this residual phenomenon seems likely to be reconciled with known laws of nature.
[476] *Proceedings of the Manchester Literary and Philosophical Society*, 28th November, 1871, vol. xi. p. 33. Since the above remarks were written, Professor Balfour Stewart has pointed out to me his paper in the *Proceedings of the Manchester Literary and Philosophical Society* for 15th November, 1870 (vol. x. p. 32), in which he shows that a body moving in an enclosure of uniform temperature would probably experience resistance independently of the presence of a ponderable medium, such as gas, between the moving body and the enclosure. The proof is founded on the theory of the dissipation of energy, and this view is said to be accepted by Professors Thomson and Tait. The enclosure is used in this case by Professor Stewart simply as a means of obtaining a proof, just as it was used by him on a previous occasion to obtain a proof of certain consequences of the Theory of Exchanges. He is of opinion that in both of these cases when once the proof has been obtained, the enclosure may be dispensed with. We know, for instance, that the relation between the inductive and absorptive powers of bodies--although this relation may have been proved by means of an enclosure, does not depend upon its presence, and Professor Stewart thinks that in like manner two bodies, or at least two bodies possessing heat such as the sun and the earth in motion relative to each other, will have the differential motion retarded until perhaps it is ultimately destroyed.
In other cases residual phenomena have involved important inferences not recognised at the time. Newton showed how the velocity of sound in the atmosphere could be calculated by a theory of pulses or undulations from the observed tension and density of the air. He inferred that the velocity in the ordinary state of the atmosphere at the earth’s surface would be 968 feet per second, and rude experiments made by him in the cloisters of Trinity College seemed to show that this was not far from the truth. Subsequently it was ascertained by other experimentalists that the velocity of sound was more nearly 1,142 feet, and the discrepancy being one-sixth part of the whole was far too much to attribute to casual errors in the numerical data. Newton attempted to explain away this discrepancy by hypotheses as to the reactions of the molecules of air, but without success.
New investigations having been made from time to time concerning the velocity of sound, both as observed experimentally and as calculated from theory, it was found that each of Newton’s results was inaccurate, the theoretical velocity being 916 feet per second, and the real velocity about 1,090 feet. The discrepancy, nevertheless, remained as serious as ever, and it was not until the year 1816 that Laplace showed it to be due to the heat developed by the sudden compression of the air in the passage of the wave, this heat having the effect of increasing the elasticity of the air and accelerating the impulse. It is now perceived that this discrepancy really involves the doctrine of the equivalence of heat and energy, and it was applied by Mayer, at least by implication, to give an estimate of the mechanical equivalent of heat. The estimate thus derived agrees satisfactorily with direct determinations by Dr. Joule and other physicists, so that the explanation of the residual phenomenon which exercised Newton’s ingenuity is now complete, and forms an important part of the new science of thermodynamics.
As Herschel observed, almost all great astronomical discoveries have been disclosed in the form of residual differences. It is the practice at well-conducted observatories to compare the positions of the heavenly bodies as actually observed with what might have been expected theoretically. This practice was introduced by Halley when Astronomer Royal, and his reduction of the lunar observations gave a series of residual errors from 1722 to 1739, by the examination of which the lunar theory was improved. Most of the greater astronomical variations arising from nutation, aberration, planetary perturbation were discovered in the same manner. The precession of the equinox was perhaps the earliest residual difference observed; the systematic divergence of Uranus from its calculated places was one of the latest, and was the clue to the remarkable discovery of Neptune. We may also class under residual phenomena all the so-called *proper motions* of the stars. A complete star catalogue, such as that of the British Association, gives a greater or less amount of proper motion for almost every star, consisting in the apparent difference of position of the star as derived from the earliest and latest good observations. But these apparent motions are often due, as explained by Baily,[477] the author of the catalogue, to errors of observation and reduction. In many cases the best astronomical authorities have differed as to the very direction of the supposed proper motion of stars, and as regards the amount of the motion, for instance of α Polaris, the most different estimates have been formed. Residual quantities will often be so small that their very existence is doubtful. Only the gradual progress of theory and of measurement will show clearly whether a discrepancy is to be referred to casual errors of observation or to some new phenomenon. But nothing is more requisite for the progress of science than the careful recording and investigation of such discrepancies. In no part of physical science can we be free from exceptions and outstanding facts, of which our present knowledge can give no account. It is among such anomalies that we must look for the clues to new realms of facts worthy of discovery. They are like the floating waifs which led Columbus to suspect the existence of the new world.
[477] *British Association Catalogue of Stars*, p. 49.Return to passage
XXVI. CHARACTER OF THE EXPERIMENTALIST
In the present age there seems to be a tendency to believe that the importance of individual genius is less than it was--
“The individual withers, and the world is more and more.”
Society, it is supposed, has now assumed so highly developed a form, that what was accomplished in past times by the solitary exertions of a great intellect, may now be worked out by the united labours of an army of investigators. Just as the well-organised power of a modern army supersedes the single-handed bravery of the mediæval knights, so we are to believe that the combination of intellectual labour has superseded the genius of an Archimedes, a Newton, or a Laplace. So-called original research is now regarded as a profession, adopted by hundreds of men, and communicated by a system of training. All that we need to secure additions to our knowledge of nature is the erection of great laboratories, museums, and observatories, and the offering of pecuniary rewards to those who can invent new chemical compounds, detect new species, or discover new comets. Doubtless this is not the real meaning of the eminent men who are now urging upon Government the endowment of physical research. They can only mean that the greater the pecuniary and material assistance given to men of science, the greater the result which the available genius of the country may be expected to produce. Money and opportunities of study can no more produce genius than sunshine and moisture can generate living beings; the inexplicable germ is wanting in both cases. But as, when the germ is present, the plant will grow more or less vigorously according to the circumstances in which it is placed, so it may be allowed that pecuniary assistance may favour development of intellect. Public opinion however is not discriminating, and is likely to interpret the agitation for the endowment of science as meaning that science can be had for money.
All such notions are erroneous. In no branch of human affairs, neither in politics, war, literature, industry, nor science, is the influence of genius less considerable than it was. It is possible that the extension and organisation of scientific study, assisted by the printing-press and the accelerated means of communication, has increased the rapidity with which new discoveries are made known, and their details worked out by many heads and hands. A Darwin now no sooner propounds original ideas concerning the evolution of living creatures, than those ideas are discussed and illustrated, and applied by naturalists in every part of the world. In former days his discoveries would have been hidden for decades of years in scarce manuscripts, and generations would have passed away before his theory had enjoyed the same amount of criticism and corroboration as it has already received. The result is that the genius of Darwin is more valuable, not less valuable, than it would formerly have been. The advance of military science and the organisation of enormous armies has not decreased the value of a skilful general; on the contrary, the rank and file are still more in need than they used to be of the guiding power of a far-seeing intellect. The swift destruction of the French military power was not due alone to the perfection of the German army, nor to the genius of Moltke; it was due to the combination of a well-disciplined multitude with a leader of the highest powers. So in every branch of human affairs the influence of the individual is not withering, but is growing with the extent of the material resources which are at his command.
Turning to our own subject, it is a work of undiminished interest to reflect upon those qualities of mind which lead to great advances in natural knowledge. Nothing, indeed, is less amenable than genius to scientific analysis and explanation. Even definition is out of the question. Buffon said that “genius is patience,” and certainly patience is one of its most requisite components. But no one can suppose that patient labour alone will invariably lead to those conspicuous results which we attribute to genius. In every branch of science, literature, art, or industry, there are thousands of men and women who work with unceasing patience, and thereby ensure moderate success; but it would be absurd to suppose that equal amounts of intellectual labour yield equal results. A Newton may modestly attribute his discoveries to industry and patient thought, and there is reason to believe that genius is unconscious and unable to account for its own peculiar powers. As genius is essentially creative, and consists in divergence from the ordinary grooves of thought and action, it must necessarily be a phenomenon beyond the domain of the laws of nature. Nevertheless, it is always an interesting and instructive work to trace out, as far as possible, the characteristics of mind by which great discoveries have been achieved, and we shall find in the analysis much to illustrate the principles of scientific method.
*Error of the Baconian Method.*
Hundreds of investigators may be constantly engaged in experimental inquiry; they may compile numberless note-books full of scientific facts, and endless tables of numerical results; but, if the views of induction here maintained be true, they can never by such work alone rise to new and great discoveries. By a system of research they may work out deductively the details of a previous discovery, but to arrive at a new principle of nature is another matter. Francis Bacon spread abroad the notion that to advance science we must begin by accumulating facts, and then draw from them, by a process of digestion, successive laws of higher and higher generality. In protesting against the false method of the scholastic logicians, he exaggerated a partially true philosophy, until it became as false as that which preceded it. His notion of scientific method was a kind of scientific bookkeeping. Facts were to be indiscriminately gathered from every source, and posted in a ledger, from which would emerge in time a balance of truth. It is difficult to imagine a less likely way of arriving at great discoveries. The greater the array of facts, the less is the probability that they will by any routine system of classification disclose the laws of nature they embody. Exhaustive classification in all possible orders is out of the question, because the possible orders are practically infinite in number.
It is before the glance of the philosophic mind that facts must display their meaning, and fall into logical order. The natural philosopher must therefore have, in the first place, a mind of impressionable character, which is affected by the slightest exceptional phenomenon. His associating and identifying powers must be great, that is, a strange fact must suggest to his mind whatever of like nature has previously come within his experience. His imagination must be active, and bring before his mind multitudes of relations in which the unexplained facts may possibly stand with regard to each other, or to more common facts. Sure and vigorous powers of deductive reasoning must then come into play, and enable him to infer what will happen under each supposed condition. Lastly, and above all, there must be the love of certainty leading him diligently and with perfect candour, to compare his speculations with the test of fact and experiment.
*Freedom of Theorising.*
It would be an error to suppose that the great discoverer seizes at once upon the truth, or has any unerring method of divining it. In all probability the errors of the great mind exceed in number those of the less vigorous one. Fertility of imagination and abundance of guesses at truth are among the first requisites of discovery; but the erroneous guesses must be many times as numerous as those which prove well founded. The weakest analogies, the most whimsical notions, the most apparently absurd theories, may pass through the teeming brain, and no record remain of more than the hundredth part. There is nothing really absurd except that which proves contrary to logic and experience. The truest theories involve suppositions which are inconceivable, and no limit can really be placed to the freedom of hypothesis.
Kepler is an extraordinary instance to this effect. No minor laws of nature are more firmly established than those which he detected concerning the orbits and motions of planetary masses, and on these empirical laws the theory of gravitation was founded. Did we not learn from his own writings the multitude of errors into which he fell, we might have imagined that he had some special faculty of seizing on the truth. But, as is well known, he was full of chimerical notions; his favourite and long-studied theory was founded on a fanciful analogy between the planetary orbits and the regular solids. His celebrated laws were the outcome of a lifetime of speculation, for the most part vain and groundless. We know this because he had a curious pleasure in dwelling upon erroneous and futile trains of reasoning, which most persons consign to oblivion. But Kepler’s name was destined to be immortal, on account of the patience with which he submitted his hypotheses to comparison with observation, the candour with which he acknowledged failure after failure, and the perseverance and ingenuity with which he renewed his attack upon the riddles of nature.
Next after Kepler perhaps Faraday is the physical philosopher who has given us the best insight into the progress of discovery, by recording erroneous as well as successful speculations. The recorded notions, indeed, are probably but a tithe of the fancies which arose in his active brain. As Faraday himself said--“The world little knows how many of the thoughts and theories which have passed through the mind of a scientific investigator, have been crushed in silence and secrecy by his own severe criticism and adverse examination; that in the most successful instances not a tenth of the suggestions, the hopes, the wishes, the preliminary conclusions have been realised.”
Nevertheless, in Faraday’s researches, published in the *Philosophical Transactions*, in minor papers, in manuscript note-books, or in other materials, made known in his interesting life by Dr. Bence Jones, we find invaluable lessons for the experimentalist. These writings are full of speculations which we must not judge by the light of subsequent discovery. It may perhaps be said that Faraday committed to the printing press crude ideas which a friend would have counselled him to keep back. There was occasionally even a wildness and vagueness in his notions, which in a less careful experimentalist would have been fatal to the attainment of truth. This is especially apparent in a curious paper concerning Ray-vibrations; but fortunately Faraday was aware of the shadowy character of his speculations, and expressed the feeling in words which must be quoted. “I think it likely,” he says,[478] “that I have made many mistakes in the preceding pages, for even to myself my ideas on this point appear only as the shadow of a speculation, or as one of those impressions upon the mind, which are allowable for a time as guides to thought and research. He who labours in experimental inquiries knows how numerous these are, and how often their apparent fitness and beauty vanish before the progress and development of real natural truth.” If, then, the experimentalist has no royal road to the discovery of the truth, it is an interesting matter to consider by what logical procedure he attains the truth.
[478] *Experimental Researches in Chemistry and Physics*, p. 372. *Philosophical Magazine*, 3rd Series, May 1846, vol. xxviii. p. 350.
If I have taken a correct view of logical method, there is really no such thing as a distinct process of induction. The probability is infinitely small that a collection of complicated facts will fall into an arrangement capable of exhibiting directly the laws obeyed by them. The mathematician might as well expect to integrate his functions by a ballot-box, as the experimentalist to draw deep truths from haphazard trials. All induction is but the inverse application of deduction, and it is by the inexplicable action of a gifted mind that a multitude of heterogeneous facts are ranged in luminous order as the results of some uniformly acting law. So different, indeed, are the qualities of mind required in different branches of science, that it would be absurd to attempt to give an exhaustive description of the character of mind which leads to discovery. The labours of Newton could not have been accomplished except by a mind of the utmost mathematical genius; Faraday, on the other hand, has made the most extensive additions to human knowledge without passing beyond common arithmetic. I do not remember meeting in Faraday’s writings with a single algebraic formula or mathematical problem of any complexity. Professor Clerk Maxwell, indeed, in the preface to his new *Treatise on Electricity*, has strongly recommended the reading of Faraday’s researches by all students of science, and has given his opinion that though Faraday seldom or never employed mathematical formulæ, his methods and conceptions were not the less mathematical in their nature.[479] I have myself protested against the prevailing confusion between a mathematical and an exact science,[480] yet I certainly think that Faraday’s experiments were for the most part qualitative, and that his mathematical ideas were of a rudimentary character. It is true that he could not possibly investigate such a subject as magne-crystallic action without involving himself in geometrical relations of some complexity. Nevertheless I think that he was deficient in mathematical deductive power, that power which is so highly developed by the modern system of mathematical training at Cambridge.
[479] See also *Nature*, September 18, 1873; vol. viii. p. 398.
[480] *Theory of Political Economy*, pp. 3–14.
Faraday was acquainted with the forms of his celebrated lines of force, but I am not aware that he ever entered into the algebraic nature of those curves, and I feel sure that he could not have explained their forms as depending on the resultant attractions of all the magnetic particles. There are even occasional indications that he did not understand some of the simpler mathematical doctrines of modern physical science. Although he so clearly foresaw the correlation of the physical forces, and laboured so hard with his own hands to connect gravity with other forces, it is doubtful whether he understood the doctrine of the conservation of energy as applied to gravitation. Faraday was probably equal to Newton in experimental skill, and in that peculiar kind of deductive power which leads to the invention of simple qualitative experiments; but it must be allowed that he exhibited little of that mathematical power which enabled Newton to follow out intuitively the quantitative results of a complicated problem with such wonderful facility. Two instances, Newton and Faraday, are sufficient to show that minds of widely different conformation will meet with suitable regions of research. Nevertheless, there are certain traits which we may discover in all the highest scientific minds.
*The Newtonian Method, the True Organum.*
Laplace was of opinion that the *Principia* and the *Opticks* of Newton furnished the best models then available of the delicate art of experimental and theoretical investigation. In these, as he says, we meet with the most happy illustrations of the way in which, from a series of inductions, we may rise to the causes of phenomena, and thence descend again to all the resulting details.
The popular notion concerning Newton’s discoveries is that in early life, when driven into the country by the Great Plague, a falling apple accidentally suggested to him the existence of gravitation, and that, availing himself of this hint, he was led to the discovery of the law of gravitation, the explanation of which constitutes the *Principia*. It is difficult to imagine a more ludicrous and inadequate picture of Newton’s labours. No originality, or at least priority, was claimed by Newton as regards the discovery of the law of the inverse square, so closely associated with his name. In a well-known Scholium[481] he acknowledges that Sir Christopher Wren, Hooke, and Halley, had severally observed the accordance of Kepler’s third law of motion with the principle of the inverse square.
[481] *Principia*, bk. i. Prop. iv.
Newton’s work was really that of developing the methods of deductive reasoning and experimental verification, by which alone great hypotheses can be brought to the touchstone of fact. Archimedes was the greatest of ancient philosophers, for he showed how mathematical theory could be wedded to physical experiments; and his works are the first true Organum. Newton is the modern Archimedes, and the *Principia* forms the true Novum Organum of scientific method. The laws which he established are great, but his example of the manner of establishing them is greater still. Excepting perhaps chemistry and electricity, there is hardly a progressive branch of physical and mathematical science, which has not been developed from the germs of true scientific procedure which he disclosed in the *Principia* or the *Opticks*. Overcome by the success of his theory of universal gravitation, we are apt to forget that in his theory of sound he originated the mathematical investigation of waves and the mutual action of particles; that in his corpuscular theory of light, however mistaken, he first ventured to apply mathematical calculation to molecular attractions and repulsions; that in his prismatic experiments he showed how far experimental verification could be pushed; that in his examination of the coloured rings named after him, he accomplished the most remarkable instance of minute measurement yet known, a mere practical application of which by Fizeau was recently deemed worthy of a medal by the Royal Society. We only learn by degrees how complete was his scientific insight; a few words in his third law of motion display his acquaintance with the fundamental principles of modern thermodynamics and the conservation of energy, while manuscripts long overlooked prove that in his inquiries concerning atmospheric refraction he had overcome the main difficulties of applying theory to one of the most complex of physical problems.
After all, it is only by examining the way in which he effected discoveries, that we can rightly appreciate his greatness. The *Principia* treats not of gravity so much as of forces in general, and the methods of reasoning about them. He investigates not one hypothesis only, but mechanical hypotheses in general. Nothing so much strikes the reader of the work as the exhaustiveness of his treatment, and the unbounded power of his insight. If he treats of central forces, it is not one law of force which he discusses, but many, or almost all imaginable laws, the results of each of which he sketches out in a few pregnant words. If his subject is a resisting medium, it is not air or water alone, but resisting media in general. We have a good example of his method in the scholium to the twenty-second proposition of the second book, in which he runs rapidly over many suppositions as to the laws of the compressing forces which might conceivably act in an atmosphere of gas, a consequence being drawn from each case, and that one hypothesis ultimately selected which yields results agreeing with experiments upon the pressure and density of the terrestrial atmosphere.
Newton said that he did not frame hypotheses, but, in reality, the greater part of the *Principia* is purely hypothetical, endless varieties of causes and laws being imagined which have no counterpart in nature. The most grotesque hypotheses of Kepler or Descartes were not more imaginary. But Newton’s comprehension of logical method was perfect; no hypothesis was entertained unless it was definite in conditions, and admitted of unquestionable deductive reasoning; and the value of each hypothesis was entirely decided by the comparison of its consequences with facts. I do not entertain a doubt that the general course of his procedure is identical with that view of the nature of induction, as the inverse application of deduction, which I advocate throughout this book. Francis Bacon held that science should be founded on experience, but he mistook the true mode of using experience, and, in attempting to apply his method, ludicrously failed. Newton did not less found his method on experience, but he seized the true method of treating it, and applied it with a power and success never since equalled. It is a great mistake to say that modern science is the result of the Baconian philosophy; it is the Newtonian philosophy and the Newtonian method which have led to all the great triumphs of physical science, and I repeat that the *Principia* forms the true “Novum Organum.”
In bringing his theories to a decisive experimental verification, Newton showed, as a general rule, exquisite skill and ingenuity. In his hands a few simple pieces of apparatus were made to give results involving an unsuspected depth of meaning. His most beautiful experimental inquiry was that by which he proved the differing refrangibility of rays of light. To suppose that he originally discovered the power of a prism to break up a beam of white light would be a mistake, for he speaks of procuring a glass prism to try the “celebrated phenomena of colours.” But we certainly owe to him the theory that white light is a mixture of rays differing in refrangibility, and that lights which differ in colour, differ also in refrangibility. Other persons might have conceived this theory; in fact, any person regarding refraction as a quantitative effect must see that different parts of the spectrum have suffered different amounts of refraction. But the power of Newton is shown in the tenacity with which he followed his theory into every consequence, and tested each result by a simple but conclusive experiment. He first shows that different coloured spots are displaced by different amounts when viewed through a prism, and that their images come to a focus at different distances from the lense, as they should do, if the refrangibility differed. After excluding by many experiments a variety of indifferent circumstances, he fixes his attention upon the question whether the rays are merely shattered, disturbed, and spread out in a chance manner, as Grimaldi supposed, or whether there is a constant relation between the colour and the refrangibility.
If Grimaldi was right, it might be expected that a part of the spectrum taken separately, and subjected to a second refraction, would suffer a new breaking up, and produce some new spectrum. Newton inferred from his own theory that a particular ray of the spectrum would have a constant refrangibility, so that a second prism would merely bend it more or less, but not further disperse it in any considerable degree. By simply cutting off most of the rays of the spectrum by a screen, and allowing the remaining narrow ray to fall on a second prism, he proved the truth of this conclusion; and then slowly turning the first prism, so as to vary the colour of the ray falling on the second one, he found that the spot of light formed by the twice-refracted ray travelled up and down, a palpable proof that the amount of refrangibility varies with the colour. For his further satisfaction, he sometimes refracted the light a third or fourth time, and he found that it might be refracted upwards or downwards or sideways, and yet for each colour there was a definite amount of refraction through each prism. He completed the proof by showing that the separated rays may again be gathered together into white light by an inverted prism, so that no number of refractions alters the character of the light. The conclusion thus obtained serves to explain the confusion arising in the use of a common lense; he shows that with homogeneous light there is one distinct focus, with mixed light an infinite number of foci, which prevent a clear view from being obtained at any point.
What astonishes the reader of the *Opticks* is the persistence with which Newton follows out the consequences of a preconceived theory, and tests the one notion by a wonderful variety of simple comparisons with fact. The ease with which he invents new combinations, and foresees the results, subsequently verified, produces an insuperable conviction in the reader that he has possession of the truth. And it is certainly the theory which leads him to the experiments, most of which could hardly be devised by accident. Newton actually remarks that it was by mathematically determining all kinds of phenomena of colours which could be produced by refraction that he had “invented” almost all the experiments in the book, and he promises that others who shall “argue truly,” and try the experiments with care, will not be disappointed in the results.[482]
[482] *Opticks*, bk. i. part ii. Prop. 3. 3rd ed. p. 115.
The philosophic method of Huyghens was the same as that of Newton, and Huyghens’ investigation of double refraction furnishes almost equally beautiful instances of theory guiding experiment. So far as we know double refraction was first discovered by accident, and was described by Erasmus Bartholinus in 1669. The phenomenon then appeared to be entirely exceptional, and the laws governing the two paths of the refracted rays were so unapparent and complicated, that Newton altogether misunderstood the phenomenon, and it was only at the latter end of the last century that scientific men began to comprehend its laws.
Nevertheless, Huyghens had, with rare genius, arrived at the true theory as early as 1678. He regarded light as an undulatory motion of some medium, and in his *Traité de la Lumière* he pointed out that, in ordinary refraction, the velocity of propagation of the wave is equal in all directions, so that the front of an advancing wave is spherical, and reaches equal distances in equal times. But in crystals, as he supposed, the medium would be of unequal elasticity in different directions, so that a disturbance would reach unequal distances in equal times, and the wave produced would have a spheroidal form. Huyghens was not satisfied with an unverified theory. He calculated what might be expected to happen when a crystal of calc-spar was cut in various directions, and he says: “I have examined in detail the properties of the extraordinary refraction of this crystal, to see if each phenomenon which is deduced from theory would agree with what is really observed. And this being so, it is no slight proof of the truth of our suppositions and principles; but what I am going to add here confirms them still more wonderfully; that is, the different modes of cutting this crystal, in which the surfaces produced give rise to refraction exactly such as they ought to be, and as I had foreseen them, according to the preceding theory.”
Newton’s mistaken corpuscular theory of light caused the theories and experiments of Huyghens to be disregarded for more than a century; but it is not easy to imagine a more beautiful or successful application of the true method of inductive investigation, theory guiding experiment, and yet wholly relying on experiment for confirmation.
*Candour and Courage of the Philosophic Mind.*
Perfect readiness to reject a theory inconsistent with fact is a primary requisite of the philosophic mind. But it would be a mistake to suppose that this candour has anything akin to fickleness; on the contrary, readiness to reject a false theory may be combined with a peculiar pertinacity and courage in maintaining an hypothesis as long as its falsity is not actually apparent. There must, indeed, be no prejudice or bias distorting the mind, and causing it to pass over the unwelcome results of experiment. There must be that scrupulous honesty and flexibility of mind, which assigns adequate value to all evidence; indeed, the more a man loves his theory, the more scrupulous should be his attention to its faults. It is common in life to meet with some theorist, who, by long cogitation over a single theory, has allowed it to mould his mind, and render him incapable of receiving anything but as a contribution to the truth of his one theory. A narrow and intense course of thought may sometimes lead to great results, but the adoption of a wrong theory at the outset is in such a mind irretrievable. The man of one idea has but a single chance of truth. The fertile discoverer, on the contrary, chooses between many theories, and is never wedded to any one, unless impartial and repeated comparison has convinced him of its validity. He does not choose and then compare; but he compares time after time, and then chooses.
Having once deliberately chosen, the philosopher may rightly entertain his theory with the strongest fidelity. He will neglect no objection; for he may chance at any time to meet a fatal one; but he will bear in mind the inconsiderable powers of the human mind compared with the tasks it has to undertake. He will see that no theory can at first be reconciled with all objections, because there may be many interfering causes, and the very consequences of the theory may have a complexity which prolonged investigation by successive generations of men may not exhaust. If, then, a theory exhibit a number of striking coincidences with fact, it must not be thrown aside until at least one *conclusive discordance* is proved, regard being had to possible error in establishing that discordance. In science and philosophy something must be risked. He who quails at the least difficulty will never establish a new truth, and it was not unphilosophic in Leslie to remark concerning his own inquiries into the nature of heat--
“In the course of investigation, I have found myself compelled to relinquish some preconceived notions; but I have not abandoned them hastily, nor, till after a warm and obstinate defence, I was driven from every post.”[483]
[483] *Experimental Inquiry into the Nature of Heat.* Preface, p. xv.
Faraday’s life, again, furnishes most interesting illustrations of this tenacity of the philosophic mind. Though so candid in rejecting some theories, there were others to which he clung through everything. One of his favourite notions resulted in a brilliant discovery; another remains in doubt to the present day.
*The Philosophic Character of Faraday.*
In Faraday’s researches concerning the connection of magnetism and light, we find an excellent instance of the pertinacity with which a favourite theory may be pursued, so long as the results of experiment do not clearly negative the notions entertained. In purely quantitative questions, as we have seen, the absence of apparent effect can seldom be regarded as proving the absence of all effect. Now Faraday was convinced that some mutual relation must exist between magnetism and light. As early as 1822, he attempted to produce an effect upon a ray of polarised light, by passing it through water placed between the poles of a voltaic battery; but he was obliged to record that not the slightest effect was observable. During many years the subject, we are told,[484] rose again and again to his mind, and no failure could make him relinquish his search after this unknown relation. It was in the year 1845 that he gained the first success; on August 30th he began to work with common electricity, vainly trying glass, quartz, Iceland spar, &c. Several days of labour gave no result; yet he did not desist. Heavy glass, a transparent medium of great refractive powers, composed of borate of lead, was now tried, being placed between the poles of a powerful electro-magnet while a ray of polarised light was transmitted through it. When the poles of the electro-magnet were arranged in certain positions with regard to the substance under trial, no effects were apparent; but at last Faraday happened fortunately to place a piece of heavy glass so that contrary magnetic poles were on the same side, and now an effect was witnessed. The glass was found to have the power of twisting the plane of polarisation of the ray of light.
[484] Bence Jones, *Life of Faraday*, vol. i. p. 362.
All Faraday’s recorded thoughts upon this great experiment are replete with curious interest. He attributes his success to the opinion, almost amounting to a conviction, that the various forms, under which the forces of matter are made manifest, have one common origin, and are so directly related and mutually dependent that they are convertible. “This strong persuasion,” he says,[485] “extended to the powers of light, and led to many exertions having for their object the discovery of the direct relation of light and electricity. These ineffectual exertions could not remove my strong persuasion, and I have at last succeeded.” He describes the phenomenon in somewhat figurative language as *the magnetisation of a ray of light*, and also as *the illumination of a magnetic curve or line of force*. He has no sooner got the effect in one case, than he proceeds, with his characteristic comprehensiveness of research, to test the existence of a like phenomenon in all the substances available. He finds that not only heavy glass, but solids and liquids, acids and alkalis, oils, water, alcohol, ether, all possess this power; but he was not able to detect its existence in any gaseous substance. His thoughts cannot be restrained from running into curious speculations as to the possible results of the power in certain cases. “What effect,” he says, “does this force have in the earth where the magnetic curves of the earth traverse its substance? Also what effect in a magnet?” And then he falls upon the strange notion that perhaps this force tends to make iron and oxide of iron transparent, a phenomenon never observed. We can meet with nothing more instructive as to the course of mind by which great discoveries are made, than these records of Faraday’s patient labours, and his varied success and failure. Nor are his unsuccessful experiments upon the relation of gravity and electricity less interesting, or less worthy of study.
[485] Ibid. vol. ii. p. 199.
Throughout a large part of his life, Faraday was possessed by the idea that gravity cannot be unconnected with the other forces of nature. On March 19th, 1849, he wrote in his laboratory book,--“Gravity. Surely this force must be capable of an experimental relation to electricity, magnetism, and the other forces, so as to bind it up with them in reciprocal action and equivalent effect?”[486] He filled twenty paragraphs or more with reflections and suggestions, as to the mode of treating the subject by experiment. He anticipated that the mutual approach of two bodies would develop electricity in them, or that a body falling through a conducting helix would excite a current changing in direction as the motion was reversed. “*All this is a dream*,” he remarks; “still examine it by a few experiments. Nothing is too wonderful to be true, if it be consistent with the laws of nature; and in such things as these, experiment is the best test of such consistency.”
[486] See also his more formal statement in the *Experimental Researches in Electricity*, 24th Series, § 2702, vol. iii. p. 161.
He executed many difficult and tedious experiments, which are described in the 24th Series of Experimental Researches. The result was *nil*, and yet he concludes: “Here end my trials for the present. The results are negative; they do not shake my strong feeling of the existence of a relation between gravity and electricity, though they give no proof that such a relation exists.”
He returned to the work when he was ten years older, and in 1858–9 recorded many remarkable reflections and experiments. He was much struck by the fact that electricity is essentially a *dual force*, and it had always been a conviction of Faraday that no body could be electrified positively without some other body becoming electrified negatively; some of his researches had been simple developments of this relation. But observing that between two mutually gravitating bodies there was no apparent circumstance to determine which should be positive and which negative, he does not hesitate to call in question an old opinion. “The evolution of *one* electricity would be a new and very remarkable thing. The idea throws a doubt on the whole; but still try, for who knows what is possible in dealing with gravity?” We cannot but notice the candour with which he thus acknowledges in his laboratory book the doubtfulness of the whole thing, and is yet prepared as a forlorn hope to frame experiments in opposition to all his previous experience of the course of nature. For a time his thoughts flow on as if the strange detection were already made, and he had only to trace out its consequences throughout the universe. “Let us encourage ourselves by a little more imagination prior to experiment,” he says; and then he reflects upon the infinity of actions in nature, in which the mutual relations of electricity and gravity would come into play; he pictures to himself the planets and the comets charging themselves as they approach the sun; cascades, rain, rising vapour, circulating currents of the atmosphere, the fumes of a volcano, the smoke in a chimney become so many electrical machines. A multitude of events and changes in the atmosphere seem to be at once elucidated by such actions; for a moment his reveries have the vividness of fact. “I think we have been dull and blind not to have suspected some such results,” and he sums up rapidly the consequences of his great but imaginary theory; an entirely new mode of exciting heat or electricity, an entirely new relation of the natural forces, an analysis of gravitation, and a justification of the conservation of force.
Such were Faraday’s fondest dreams of what might be, and to many a philosopher they would have been sufficient basis for the writing of a great book. But Faraday’s imagination was within his full control; as he himself says, “Let the imagination go, guarding it by judgment and principle, and holding it in and directing it by experiment.” His dreams soon took a very practical form, and for many days he laboured with ceaseless energy, on the staircase of the Royal Institution, in the clock tower of the Houses of Parliament, or at the top of the Shot Tower in Southwark, raising and lowering heavy weights, and combining electrical helices and wires in every conceivable way. His skill and long experience in experiment were severely taxed to eliminate the effects of the earth’s magnetism, and time after time he saved himself from accepting mistaken indications, which to another man might have seemed conclusive verifications of his theory. When all was done there remained absolutely no results. “The experiments,” he says, “were well made, but the results are negative;” and yet, he adds, “I cannot accept them as conclusive.” In this position the question remains to the present day; it may be that the effect was too slight to be detected, or it may be that the arrangements adopted were not suited to develop the particular relation which exists, just as Oersted could not detect electro-magnetism, so long as his wire was perpendicular to the plane of motion of his needle. But these are not matters which concern us further here. We have only to notice the profound conviction in the unity of natural laws, the active powers of inference and imagination, the unbounded licence of theorising, combined above all with the utmost diligence in experimental verification which this remarkable research exhibits.
*Reservation of Judgment.*
There is yet another characteristic needed in the philosophic mind; it is that of suspending judgment when the data are insufficient. Many people will express a confident opinion on almost any question which is put before them, but they thereby manifest not strength, but narrowness of mind. To see all sides of a complicated subject, and to weigh all the different facts and probabilities correctly, require no ordinary powers of comprehension. Hence it is most frequently the philosophic mind which is in doubt, and the ignorant mind which is ready with a positive decision. Faraday has himself said, in a very interesting lecture:[487] “Occasionally and frequently the exercise of the judgment ought to end in *absolute reservation*. It may be very distasteful, and great fatigue, to suspend a conclusion; but as we are not infallible, so we ought to be cautious; we shall eventually find our advantage, for the man who rests in his position is not so far from right as he who, proceeding in a wrong direction, is ever increasing his distance.”
[487] Printed in *Modern Culture*, edited by Youmans, p. 219.
Arago presented a conspicuous example of this high quality of mind, as Faraday remarks; for when he made known his curious discovery of the relation of a magnetic needle to a revolving copper plate, a number of supposed men of science in different countries gave immediate and confident explanations of it, which were all wrong. But Arago, who had both discovered the phenomenon and personally investigated its conditions, declined to put forward publicly any theory at all.
At the same time we must not suppose that the truly philosophic mind can tolerate a state of doubt, while a chance of decision remains open. In science nothing like compromise is possible, and truth must be one. Hence, doubt is the confession of ignorance, and involves a painful feeling of incapacity. But doubt lies between error and truth, so that if we choose wrongly we are further away than ever from our goal.
Summing up, then, it would seem as if the mind of the great discoverer must combine contradictory attributes. He must be fertile in theories and hypotheses, and yet full of facts and precise results of experience. He must entertain the feeblest analogies, and the merest guesses at truth, and yet he must hold them as worthless till they are verified in experiment. When there are any grounds of probability he must hold tenaciously to an old opinion, and yet he must be prepared at any moment to relinquish it when a clearly contradictory fact is encountered. “The philosopher,” says Faraday,[488] “should be a man willing to listen to every suggestion, but determined to judge for himself. He should not be biased by appearances; have no favourite hypothesis; be of no school; and in doctrine have no master. He should not be a respecter of persons, but of things. Truth should be his primary object. If to these qualities be added industry, he may indeed hope to walk within the veil of the temple of nature.”
[488] *Life of Faraday*, vol. i. p. 225.Return to passage
XXVII. GENERALISATION
I have endeavoured to show in preceding chapters that all inductive reasoning is an inverse application of deductive reasoning, and consists in demonstrating that the consequences of certain assumed laws agree with facts of nature gathered by active or passive observation. The fundamental process of reasoning, as stated in the outset, consists in inferring of a thing what we know of similar objects, and it is on this principle that the whole of deductive reasoning, whether simply logical or mathematico-logical, is founded. All inductive reasoning must be founded on the same principle. It might seem that by a plain use of this principle we could avoid the complicated processes of induction and deduction, and argue directly from one particular case to another, as Mill proposed. If the Earth, Venus, Mars, Jupiter, and other planets move in elliptic orbits, cannot we dispense with elaborate precautions, and assert that Neptune, Ceres, and the last discovered planet must do so likewise? Do we not know that Mr. Gladstone must die, because he is like other men? May we not argue that because some men die therefore he must? Is it requisite to ascend by induction to the general proposition “all men must die,” and then descend by deduction from that general proposition to the case of Mr. Gladstone? My answer undoubtedly is that we must ascend to general propositions. The fundamental principle of the substitution of similars gives us no warrant in affirming of Mr. Gladstone what we know of other men, because we cannot be sure that Mr. Gladstone is exactly similar to other men. Until his death we cannot be perfectly sure that he possesses all the attributes of other men; it is a question of probability, and I have endeavoured to explain the mode in which the theory of probability is applied to calculate the probability that from a series of similar events we may infer the recurrence of like events under identical circumstances. There is then no such process as that of inferring from particulars to particulars. A careful analysis of the conditions under which such an inference appears to be made, shows that the process is really a general one, and that what is inferred of a particular case might be inferred of all similar cases. All reasoning is essentially general, and all science implies generalisation. In the very birth-time of philosophy this was held to be so: “Nulla scientia est de individuis, sed de solis universalibus,” was the doctrine of Plato, delivered by Porphyry. And Aristotle[489] held a like opinion--Οὐδεμία δὲ τέχνη σκοπεȋ τὸ καθ’ ἕκαστον ... τὸ δὲ καθ’ ἕκαστον ἄπειρον καὶ οὐκ ἐπιστητόν. “No art treats of particular cases; for particulars are infinite and cannot be known.” No one who holds the doctrine that reasoning may be from particulars to particulars, can be supposed to have the most rudimentary notion of what constitutes reasoning and scíence.
[489] Aristotle’s *Rhetoric*, Liber I. 2. 11.
At the same time there can be no doubt that practically what we find to be true of many similar objects will probably be true of the next similar object. This is the result to which an analysis of the Inverse Method of Probabilities leads us, and, in the absence of precise data from which we may calculate probabilities, we are usually obliged to make a rough assumption that similars in some respects are similars in other respects. Thus it comes to pass that a large part of the reasoning processes in which scientific men are engaged, consists in detecting similarities between objects, and then rudely assuming that the like similarities will be detected in other cases.
*Distinction of Generalisation and Analogy.*
There is no distinction but that of degree between what is known as reasoning by *generalisation* and reasoning by *analogy*. In both cases from certain observed resemblances we infer, with more or less probability, the existence of other resemblances. In generalisation the resemblances have great extension and usually little intension, whereas in analogy we rely upon the great intension, the extension being of small amount (p. 26). If we find that the qualities A and B are associated together in a great many instances, and have never been found separate, it is highly probable that on the next occasion when we meet with A, B will also be present, and *vice versâ*. Thus wherever we meet with an object possessing gravity, it is found to possess inertia also, nor have we met with any material objects possessing inertia without discovering that they also possess gravity. The probability has therefore become very great, as indicated by the rules founded on the Inverse Method of Probabilities (p. 257), that whenever in the future we meet an object possessing either of the properties of gravity and inertia, it will be found on examination to possess the other of these properties. This is a clear instance of the employment of generalisation.
In analogy, on the other hand, we reason from likeness in many points to likeness in other points. The qualities or points of resemblance are now numerous, not the objects. At the poles of Mars are two white spots which resemble in many respects the white regions of ice and snow at the poles of the earth. There probably exist no other similar objects with which to compare these, yet the exactness of the resemblance enables us to infer, with high probability, that the spots on Mars consist of ice and snow. In short, many points of resemblance imply many more. From the appearance and behaviour of those white spots we infer that they have all the chemical and physical properties of frozen water. The inference is of course only probable, and based upon the improbability that aggregates of many qualities should be formed in a like manner in two or more cases, without being due to some uniform condition or cause.
In reasoning by analogy, then, we observe that two objects ABCDE.... and A′B′C′D′E′.... have many like qualities, as indicated by the identity of the letters, and we infer that, since the first has another quality, X, we shall discover this quality in the second case by sufficiently close examination. As Laplace says,--“Analogy is founded on the probability that similar things have causes of the same kind, and produce the same effects. The more perfect this similarity, the greater is this probability.”[490] The nature of analogical inference is aptly described in the work on Logic attributed to Kant, where the rule of ordinary induction is stated in the words, “*Eines in vielen, also in allen*,” one quality in many things, therefore in all; and the rule of analogy is “*Vieles in einem, also auch das übrige in demselben*,”[491] many (qualities) in one, therefore also the remainder in the same. It is evident that there may be intermediate cases in which, from the identity of a moderate number of objects in several properties, we may infer to other objects. Probability must rest either upon the number of instances or the depth of resemblance, or upon the occurrence of both in sufficient degrees. What there is wanting in extension must be made up by intension, and *vice versâ*.
[490] *Essai Philosophique sur les Probabilités*, p. 86.
[491] Kant’s *Logik*, § 84, Königsberg, 1800, p. 207.
*Two Meanings of Generalisation.*
The term generalisation, as commonly used, includes two processes which are of different character, but are often closely associated together. In the first place, we generalise when we recognise even in two objects a common nature. We cannot detect the slightest similarity without opening the way to inference from one case to the other. If we compare a cubical crystal with a regular octahedron, there is little apparent similarity; but, as soon as we perceive that either can be produced by the symmetrical modification of the other, we discover a groundwork of similarity in the crystals, which enables us to infer many things of one, because they are true of the other. Our knowledge of ozone took its rise from the time when the similarity of smell, attending electric sparks, strokes of lightning, and the slow combustion of phosphorus, was noticed by Schönbein. There was a time when the rainbow was an inexplicable phenomenon--a portent, like a comet, and a cause of superstitious hopes and fears. But we find the true spirit of science in Roger Bacon, who desires us to consider the objects which present the same colours as the rainbow; he mentions hexagonal crystals from Ireland and India, but he bids us not suppose that the hexagonal form is essential, for similar colours may be detected in many transparent stones. Drops of water scattered by the oar in the sun, the spray from a water-wheel, the dewdrops lying on the grass in the summer morning, all display a similar phenomenon. No sooner have we grouped together these apparently diverse instances, than we have begun to generalise, and have acquired a power of applying to one instance what we can detect of others. Even when we do not apply the knowledge gained to new objects, our comprehension of those already observed is greatly strengthened and deepened by learning to view them as particular cases of a more general property.
A second process, to which the name of generalisation is often given, consists in passing from a fact or partial law to a multitude of unexamined cases, which we believe to be subject to the same conditions. Instead of merely recognising similarity as it is brought before us, we predict its existence before our senses can detect it, so that generalisation of this kind endows us with a prophetic power of more or less probability. Having observed that many substances assume, like water and mercury, the three states of solid, liquid, and gas, and having assured ourselves by frequent trial that the greater the means we possess of heating and cooling, the more substances we can vaporise and freeze, we pass confidently in advance of fact, and assume that all substances are capable of these three forms. Such a generalisation was accepted by Lavoisier and Laplace before many of the corroborative facts now in our possession were known. The reduction of a single comet beneath the sway of gravity was considered sufficient indication that all comets obey the same power. Few persons doubted that the law of gravity extended over the whole heavens; certainly the fact that a few stars out of many millions manifest the action of gravity, is now held to be sufficient evidence of its general extension over the visible universe.
*Value of Generalisation.*
It might seem that if we know particular facts, there can be little use in connecting them together by a general law. The particulars must be more full of useful information than an abstract general statement. If we know, for instance, the properties of an ellipse, a circle, a parabola, and hyperbola, what is the use of learning all these properties over again in the general theory of curves of the second degree? If we understand the phenomena of sound and light and water-waves separately, what is the need of erecting a general theory of waves, which, after all, is inapplicable to practice until resolved again into particular cases? But, in reality, we never do obtain an adequate knowledge of particulars until we regard them as cases of the general. Not only is there a singular delight in discovering the many in the one, and the one in the many, but there is a constant interchange of light and knowledge. Properties which are unapparent in the hyperbola may be readily observed in the ellipse. Most of the complex relations which old geometers discovered in the circle will be reproduced *mutatis mutandis* in the other conic sections. The undulatory theory of light might have been unknown at the present day, had not the theory of sound supplied hints by analogy. The study of light has made known many phenomena of interference and polarisation, the existence of which had hardly been suspected in the case of sound, but which may now be sought out, and perhaps found to possess unexpected interest. The careful study of water-waves shows how waves alter in form and velocity with varying depth of water. Analogous changes may some time be detected in sound waves. Thus there is mutual interchange of aid.
“Every study of a generalisation or extension,” De Morgan has well said,[492] “gives additional power over the particular form by which the generalisation is suggested. Nobody who has ever returned to quadratic equations after the study of equations of all degrees, or who has done the like, will deny my assertion that οὐ βλέπει βλέπων may be predicated of any one who studies a branch or a case, without afterwards making it part of a larger whole. Accordingly it is always worth while to generalise, were it only to give power over the *particular*. This principle, of daily familiarity to the mathematician, is almost unknown to the logician.”
[492] *Syllabus of a Proposed System of Logic*, p. 34.
*Comparative Generality of Properties.*
Much of the value of science depends upon the knowledge which we gradually acquire of the different degrees of generality of properties and phenomena of various kinds. The use of science consists in enabling us to act with confidence, because we can foresee the result. Now this foresight must rest upon the knowledge of the powers which will come into play. That knowledge, indeed, can never be certain, because it rests upon imperfect induction, and the most confident beliefs and predictions of the physicist may be falsified. Nevertheless, if we always estimate the probability of each belief according to the due teaching of the data, and bear in mind that probability when forming our anticipations, we shall ensure the minimum of disappointment. Even when he cannot exactly apply the theory of probabilities, the physicist may acquire the habit of making judgments in general agreement with its principles and results.
Such is the constitution of nature, that the physicist learns to distinguish those properties which have wide and uniform extension, from those which vary between case and case. Not only are certain laws distinctly laid down, with their extension carefully defined, but a scientific training gives a kind of tact in judging how far other laws are likely to apply under any particular circumstances. We learn by degrees that crystals exhibit phenomena depending upon the directions of the axes of elasticity, which we must not expect in uniform solids. Liquids, compared even with non-crystalline solids, exhibit laws of far less complexity and variety; and gases assume, in many respects, an aspect of nearly complete uniformity. To trace out the branches of science in which varying degrees of generality prevail, would be an inquiry of great interest and importance; but want of space, if there were no other reason, would forbid me to attempt it, except in a very slight manner.
Gases, so far as they are really gaseous, not only have exactly the same properties in all directions of space, but one gas exactly resembles other gases in many qualities. All gases expand by heat, according to the same law, and by nearly the same amount; the specific heats of equivalent weights are equal, and the densities are exactly proportional to the atomic weights. All such gases obey the general law, that the volume multiplied by the pressure, and divided by the absolute temperature, is constant or nearly so. The laws of diffusion and transpiration are the same in all cases, and, generally speaking, all physical laws, as distinguished from chemical laws, apply equally to all gases. Even when gases differ in chemical or physical properties, the differences are minor in degree. Thus the differences of viscosity are far less marked than in the liquid and solid states. Nearly all gases, again, are colourless, the exceptions being chlorine, the vapours of iodine, bromine, and a few other substances.
Only in one single point, so far as I am aware, do gases present distinguishing marks unknown or nearly so, in the solid and liquid states. I mean as regards the light given off when incandescent. Each gas when sufficiently heated, yields its own peculiar series of rays, arising from the free vibrations of the constituent parts of the molecules. Hence the possibility of distinguishing gases by the spectroscope. But the molecules of solids and liquids appear to be continually in conflict with each other, so that only a confused *noise* of atoms is produced, instead of a definite series of luminous chords. At the same temperature, accordingly, all solids and liquids give off nearly the same rays when strongly heated, and we have in this case an exception to the greater generality of properties in gases.
Liquids are in many ways intermediate in character between gases and solids. While incapable of possessing different elasticity in different directions, and thus denuded of the rich geometrical complexity of solids, they retain the variety of density, colour degrees of transparency, great diversity in surface tension, viscosity, coefficients of expansion, compressibility, and many other properties which we observe in solids, but not for the most part in gases. Though our knowledge of the physical properties of liquids is much wanting in generality at present, there is ground to hope that by degrees laws connecting and explaining the variations may be traced out.
Solids are in every way contrasted to gases. Each solid substance has its own peculiar degree of density, hardness, compressibility, transparency, tenacity, elasticity, power of conducting heat and electricity, magnetic properties, capability of producing frictional electricity, and so forth. Even different specimens of the same kind of substance will differ widely, according to the accidental treatment received. And not only has each substance its own specific properties, but, when crystallised, its properties vary in each direction with regard to the axes of crystallisation. The velocity of radiation, the rate of conduction of heat, the coefficients of expansibility and compressibility, the thermo-electric properties, all vary in different crystallographic directions.
It is probable that many apparent differences between liquids, and even between solids, will be explained when we learn to regard them under exactly corresponding circumstances. The extreme generality of the properties of gases is in reality only true at an infinitely high temperature, when they are all equally remote from their condensing points. Now, it is found that if we compare liquids--for instance, different kinds of alcohols--not at equal temperatures, but at points equally distant from their respective boiling points, the laws and coefficients of expansion are nearly equal. The vapour-tensions of liquids also are more nearly equal, when compared at corresponding points, and the boiling-points appear in many cases to be simply related to the chemical composition. No doubt the progress of investigation will enable us to discover generality, where at present we only see variety and puzzling complexity.
In some cases substances exhibit the same physical properties in the liquid as in the solid state. Lead has a high refractive power, whether in solution, or in solid salts, crystallised or vitreous. The magnetic power of iron is conspicuous, whatever be its chemical condition; indeed, the magnetic properties of substances, though varying with temperature, seem not to be greatly affected by other physical changes. Colour, absorptive power for heat or light rays, and a few other properties are also often the same in liquids and gases. Iodine and bromine possess a deep colour whenever they are chemically uncombined. Nevertheless, we can seldom argue safely from the properties of a substance in one condition to those in another condition. Ice is an insulator, water a conductor of electricity, and the same contrast exists in most other substances. The conducting power of a liquid for electricity increases with the temperature, while that of a solid decreases. By degrees we may learn to distinguish between those properties of matter which depend upon the intimate construction of the chemical molecule, and those which depend upon the contact, conflict, mutual attraction, or other relations of distinct molecules. The properties of a substance with respect to light seem generally to depend upon the molecule; thus, the power of certain substances to cause the plane of polarisation of a ray of light to rotate, is exactly the same whatever be its degree of density, or the diluteness of the solution in which it is contained. Taken as a whole, the physical properties of substances and their quantitative laws, present a problem of infinite complexity, and centuries must elapse before any moderately complete generalisations on the subject become possible.
*Uniform Properties of all Matter.*
Some laws are held to be true of all matter in the universe absolutely, without exception, no instance to the contrary having ever been noticed. This is the case with the laws of motion, as laid down by Galileo and Newton. It is also conspicuously true of the law of universal gravitation. The rise of modern physical science may perhaps be considered as beginning at the time when Galileo showed, in opposition to the Aristotelians, that matter is equally affected by gravity, irrespective of its form, magnitude, or texture. All objects fall with equal rapidity, when disturbing causes, such as the resistance of the air, are removed or allowed for. That which was rudely demonstrated by Galileo from the leaning tower of Pisa, was proved by Newton to a high degree of approximation, in an experiment which has been mentioned (p. 443).
Newton formed two pendulums, as nearly as possible the same in outward shape and size by taking two equal round wooden boxes, and suspending them by equal threads, eleven feet long. The pendulums were therefore equally subject to the resistance of the air. He filled one box with wood, and in the centre of oscillation of the other he placed an equal weight of gold. The pendulums were then equal in weight as well as in size; and, on setting them simultaneously in motion, Newton found that they vibrated for a length of time with equal vibrations. He tried the same experiment with silver, lead, glass, sand, common salt, water, and wheat, in place of the gold, and ascertained that the motion of his pendulum was exactly the same whatever was the kind of matter inside.[493] He considered that a difference of a thousandth part would have been apparent. The reader must observe that the pendulums were made of equal weight only in order that they might suffer equal retardation from the air. The meaning of the experiment is that all substances manifest exactly equal acceleration from the force of gravity, and that therefore the inertia or resistance of matter to force, which is the only independent measure of mass known to us, is always proportional to gravity.
[493] *Principia*, bk. iii. Prop. VI. Motte’s translation, vol. ii. p. 220.
These experiments of Newton were considered conclusive up to very recent times, when certain discordances between the theory and observations of the movements of planets led Nicolai, in 1826, to suggest that the equal gravitation of different kinds of matter might not be absolutely exact. It is perfectly philosophical thus to call in question, from time to time, some of the best accepted laws. On this occasion Bessel carefully repeated the experiments of Newton with pendulums composed of ivory, glass, marble, quartz, meteoric stones, &c., but was unable to detect the least difference. This conclusion is also confirmed by the ultimate agreement of all the calculations of physical astronomy based upon it. Whether the mass of Jupiter be calculated from the motion of its own satellites, from the effect upon the small planets, Vesta, Juno, &c., or from the perturbation of Encke’s Comet, the results are closely accordant, showing that precisely the same law of gravity applies to the most different bodies which we can observe. The gravity of a body, again, appears to be entirely independent of its other physical conditions, being totally unaffected by any alteration in the temperature, density, electric or magnetic condition, or other physical properties of the substance.
One paradoxical result of the law of equal gravitation is the theorem of Torricelli, to the effect that all liquids of whatever density fall or flow with equal rapidity. If there be two equal cisterns respectively filled with mercury and water, the mercury, though thirteen times as heavy, would flow from an aperture neither more rapidly nor more slowly than the water, and the same would be true of ether, alcohol, and other liquids, allowance being made, however, for the resistance of the air, and the differing viscosities of the liquids.
In its exact equality and its perfect independence of all circumstances, except mass and distance, the force of gravity stands apart from all the other forces and phenomena of nature, and has not yet been brought into any relation with them except through the general principle of the conservation of energy. Magnetic attraction, as remarked by Newton, follows very different laws, depending upon the chemical quality and molecular structure of each particular substance.
We must remember that in saying “all matter gravitates,” we exclude from the term matter the basis of light-undulations, which is immensely more extensive in amount, and obeys in many respects the laws of mechanics. This adamantine substance appears, so far as can be ascertained, to be perfectly uniform in its properties when existing in space unoccupied by matter. Light and heat are conveyed by it with equal velocity in all directions, and in all parts of space so far as observation informs us. But the presence of gravitating matter modifies the density and mechanical properties of the so-called ether in a way which is yet quite unexplained.[494]
Leaving gravity, it is somewhat difficult to discover other laws which are equally true of all matter. Boerhaave was considered to have established that all bodies expand by heat; but not only is the expansion very different in different substances, but we now know positive exceptions. Many liquids and a few solids contract by heat at certain temperatures. There are indeed other relations of heat to matter which seem to be universal and uniform; all substances begin to give off rays of light at the same temperature, according to the law of Draper; and gases will not be an exception if sufficiently condensed, as in the experiments of Frankland. Grove considers it to be universally true that all bodies in combining produce heat; with the doubtful exception of sulphur and selenium, all solids in becoming liquids, and all liquids in becoming gases, absorb heat; but the quantities of heat absorbed vary with the chemical qualities of the matter. Carnot’s Thermodynamic Law is held to be exactly true of all matter without distinction; it expresses the fact that the amount of mechanical energy which might be theoretically obtained from a certain amount of heat energy depends only upon the change of the temperatures, so that whether an engine be worked by water, air, alcohol, ammonia, or any other substance, the result would theoretically be the same, if the boiler and condenser were maintained at similar temperatures.
[494] Professor Lovering has pointed out how obscure and uncertain the ideas of scientific men about this ether are, in his interesting Presidential Address before the American Association at Hartford, 1874. *Silliman’s Journal*, October 1874, p. 297. *Philosophical Magazine*, vol. xlviii. p. 493.
*Variable Properties of Matter.*
I have enumerated some of the few properties of matter, which are manifested in exactly the same manner by all substances, whatever be their differences of chemical or physical constitution. But by far the greater number of qualities vary in degree; substances are more or less dense, more or less transparent, more or less compressible, more or less magnetic, and so on. One common result of the progress of science is to show that qualities supposed to be entirely absent from many substances are present only in so low a degree of intensity that the means of detection were insufficient. Newton believed that most bodies were quite unaffected by the magnet; Faraday and Tyndall have rendered it very doubtful whether any substance whatever is wholly devoid of magnetism, including under that term diamagnetism. We are rapidly learning to believe that there are no substances absolutely opaque, or non-conducting, non-electric, non-elastic, non-viscous, non-compressible, insoluble, infusible, or non-volatile. All tends to become a matter of degree, or sometimes of direction. There may be some substances oppositely affected to others, as ferro-magnetic substances are oppositely affected to diamagnetics, or as substances which contract by heat are opposed to those which expand; but the tendency is certainly for every affection of one kind of matter to be represented by something similar in other kinds. On this account one of Newton’s rules of philosophising seems to lose all validity; he said, “Those qualities of bodies which are not capable of being heightened, and remitted, and which are found in all bodies on which experiment can be made, must be considered as universal qualities of all bodies.” As far as I can see, the contrary is more probable, namely, that qualities variable in degree will be found in every substance in a greater or less degree.
It is remarkable that Newton whose method of investigation was logically perfect, seemed incapable of generalising and describing his own procedure. His celebrated “Rules of Reasoning in Philosophy,” described at the commencement of the third book of the *Principia*, are of questionable truth, and still more questionable value.
*Extreme Instances of Properties.*
Although substances usually differ only in degree, great interest may attach to particular substances which manifest a property in a conspicuous and intense manner. Every branch of physical science has usually been developed from the attention forcibly drawn to some singular substance. Just as the loadstone disclosed magnetism and amber frictional electricity, so did Iceland spar show the existence of double refraction, and sulphate of quinine the phenomenon of fluorescence. When one such startling instance has drawn the attention of the scientific world, numerous less remarkable cases of the phenomenon will be detected, and it will probably prove that the property in question is actually universal to all matter. Nevertheless, the extreme instances retain their interest, partly in a historical point of view, partly because they furnish the most convenient substances for experiment.
Francis Bacon was fully aware of the value of such examples, which he called *Ostensive Instances* or Light-giving, Free and Predominant Instances. “They are those,” he says,[495] “which show the nature under investigation naked, in an exalted condition, or in the highest degree of power; freed from impediments, or at least by its strength predominating over and suppressing them.” He mentions quicksilver as an ostensive instance of weight or density, thinking it not much less dense than gold, and more remarkable than gold as joining density to liquidity. The magnet is mentioned as an ostensive instance of attraction. It would not be easy to distinguish clearly between these ostensive instances and those which he calls *Instantiae Monodicae*, or *Irregulares*, or *Heteroclitae*, under which he places whatever is extravagant in its properties or magnitude, or exhibits least similarity to other things, such as the sun and moon among the heavenly bodies, the elephant among animals, the letter *s* among letters, or the magnet among stones.[496]
[495] *Novum Organum*, bk. ii. Aphorisms, 24, 25.
[496] Ibid. Aph. 28.
In optical science great use has been made of the high dispersive power of the transparent compounds of lead, that is, the power of giving a long spectrum (p. 432). Dollond, having noticed this peculiar dispersive power in lenses made of flint glass, employed them to produce an achromatic arrangement. The element strontium presents a contrast to lead in this respect, being characterised by a remarkably low dispersive power; but I am not aware that this property has yet been turned to account.
Compounds of lead have both a high dispersive and a high refractive index, and in the latter respect they proved very useful to Faraday. Having spent much labour in preparing various kinds of optical glass, Faraday happened to form a compound of lead, silica, and boracic acid, now known as *heavy glass*, which possessed an intensely high refracting power. Many years afterwards in attempting to discover the action of magnetism upon light he failed to detect any effect, as has been already mentioned, (p. 588), until he happened to test a piece of the heavy glass. The peculiar refractive power of this medium caused the magnetic strain to be apparent, and the rotation of the plane of polarisation was discovered.
In almost every part of physical science there is some substance of powers pre-eminent for the special purpose to which it is put. Rock-salt is invaluable for its extreme diathermancy or transparency to the least refrangible rays of the spectrum. Quartz is equally valuable for its transparency, as regards the ultra-violet or most refrangible rays. Diamond is the most highly refracting substance which is at the same time transparent; were it more abundant and easily worked it would be of great optical importance. Cinnabar is distinguished by possessing a power of rotating the plane of polarisation of light, from 15 to 17 times as much as quartz. In electric experiments copper is employed for its high conducting powers and exceedingly low magnetic properties; iron is of course indispensable for its enormous magnetic powers; while bismuth holds a like place as regards its diamagnetic powers, and was of much importance in Tyndall’s decisive researches upon the polar character of the diamagnetic force.[497] In regard to magne-crystallic action the mineral cyanite is highly remarkable, being so powerfully affected by the earth’s magnetism, that, when delicately suspended, it assumes a constant position with regard to the magnetic meridian, and may almost be used like the compass needle. Sodium is distinguished by its unique light-giving powers, which are so extraordinary that probably one half of the whole number of stars in the heavens have a yellow tinge in consequence.
[497] *Philosophical Transactions* (1856) vol. cxlvi. p. 246.
It is remarkable that water, though the most common of all fluids, is distinguished in almost every respect by extreme qualities. Of all known substances water has the highest specific heat, being thus peculiarly fitted for the purpose of warming and cooling, to which it is often put. It rises by capillary attraction to a height more than twice that of any other liquid. In the state of ice it is nearly twice as dilatable by heat as any other known solid substance.[498] In proportion to its density it has a far higher surface tension than any other substance, being surpassed in absolute tension only by mercury; and it would not be difficult to extend considerably the list of its remarkable and useful properties.
[498] *Philosophical Magazine*, 4th Series, January 1870, vol. xxxix. p. 2.
Under extreme instances we may include cases of remarkably low powers or qualities. Such cases seem to correspond to what Bacon calls *Clandestine Instances*, which exhibit a given nature in the least intensity, and as it were in a rudimentary state.[499] They may often be important, he thinks, as allowing the detection of the cause of the property by difference. I may add that in some cases they may be of use in experiments. Thus hydrogen is the least dense of all known substances, and has the least atomic weight. Liquefied nitrous oxide has the lowest refractive index of all known fluids.[500] The compounds of strontium have the lowest dispersive power. It is obvious that a property of very low degree may prove as curious and valuable a phenomenon as a property of very high degree.
[499] *Novum Organum*, bk. ii. Aphorism 25.
[500] Faraday’s *Experimental Researches in Chemistry and Physics*, p. 93.
*The Detection of Continuity.*
We should bear in mind that phenomena which are in reality of a closely similar or even identical nature, may present to the senses very different appearances. Without a careful analysis of the changes which take place, we may often be in danger of widely separating facts and processes, which are actually instances of the same law. Extreme difference of degree or magnitude is a frequent cause of error. It is truly difficult at the first moment to recognise any similarity between the gradual rusting of a piece of iron, and the rapid combustion of a heap of straw. Yet Lavoisier’s chemical theory was founded upon the similarity of the oxydising process in one case and the other. We have only to divide the iron into excessively small particles to discover that it is really the more combustible of the two, and that it actually takes fire spontaneously and burns like tinder. It is the excessive slowness of the process in the case of a massive piece of iron which disguises its real character.
If Xenophon reports truly, Socrates was misled by not making sufficient allowance for extreme differences of degree and quantity. Anaxagoras held that the sun is a fire, but Socrates rejected this opinion, on the ground that we can look at a fire, but not at the sun, and that plants grow by sunshine while they are killed by fire. He also pointed out that a stone heated in a fire is not luminous, and soon cools, whereas the sun ever remains equally luminous and hot.[501] All such mistakes evidently arise from not perceiving that difference of quantity may be so extreme as to assume the appearance of difference of quality. It is the least creditable thing we know of Socrates, that after pointing out these supposed mistakes of earlier philosophers, he advised his followers not to study astronomy.
[501] *Memorabilia*, iv. 7.
Masses of matter of very different size may be expected to exhibit apparent differences of conduct, arising from the various intensity of the forces brought into play. Many persons have thought it requisite to imagine occult forces producing the suspension of the clouds, and there have even been absurd theories representing cloud particles as minute water-balloons buoyed up by the warm air within them. But we have only to take proper account of the enormous comparative resistance which the air opposes to the fall of minute particles, to see that all cloud particles are probably constantly falling through the air, but so slowly that there is no apparent effect. Mineral matter again is always regarded as inert and incapable of spontaneous movement. We are struck by astonishment on observing in a powerful microscope, that every kind of solid matter suspended in extremely minute particles in pure water, acquires an oscillatory movement, often so marked as to resemble dancing or skipping. I conceive that this movement is due to the comparatively vast intensity of chemical action when exerted upon minute particles, the effect being 5,000 or 10,000 greater in proportion to the mass than in fragments of an inch diameter (p. 406).
Much that was formerly obscure in the science of electricity arose from the extreme differences of intensity and quantity in which this form of energy manifests itself. Between the brilliant explosive discharge of a thunder-cloud and the gentle continuous current produced by two pieces of metal and some dilute acid, there is no apparent analogy whatever. It was therefore a work of great importance when Faraday demonstrated the identity of the forces in action, showing that common frictional electricity would decompose water like that from the voltaic battery. The relation of the phenomena became plain when he succeeded in showing that it would require 800,000 discharges of his large Leyden battery to decompose one single grain of water. Lightning was now seen to be electricity of excessively high tension, but extremely small quantity, the difference being somewhat analogous to that between the force of one million gallons of water falling through one foot, and one gallon of water falling through one million feet. Faraday estimated that one grain of water acting on four grains of zinc, would yield electricity enough for a great thunderstorm.
It was long believed that electrical conductors and insulators belonged to two opposed classes of substances. Between the inconceivable rapidity with which the current passes through pure copper wire, and the apparently complete manner in which it is stopped by a thin partition of gutta-percha or gum-lac, there seemed to be no resemblance. Faraday again laboured successfully to show that these were but the extreme cases of a chain of substances varying in all degrees in their powers of conduction. Even the best conductors, such as pure copper or silver, offer resistance to the electric current. The other metals have considerably higher powers of resistance, and we pass gradually down through oxides and sulphides. The best insulators, on the other hand, allow of an atomic induction which is the necessary antecedent of conduction. Hence Faraday inferred that whether we can measure the effect or not, all substances discharge electricity more or less.[502] One consequence of this doctrine must be, that every discharge of electricity produces an induced current. In the case of the common galvanic current we can readily detect the induced current in any parallel wire or other neighbouring conductor, and can separate the opposite currents which arise at the moments when the original current begins and ends. But a discharge of high tension electricity like lightning, though it certainly occupies time and has a beginning and an end, yet lasts so minute a fraction of a second, that it would be hopeless to attempt to detect and separate the two opposite induced currents, which are nearly simultaneous and exactly neutralise each other. Thus an apparent failure of analogy is explained away, and we are furnished with another instance of a phenomenon incapable of observation and yet theoretically known to exist.[503]
[502] *Experimental Researches in Electricity*, Series xii. vol. i. p. 420.
[503] *Life of Faraday*, vol. ii. p. 7.
Perhaps the most extraordinary case of the detection of unsuspected continuity is found in the discovery of Cagniard de la Tour and Professor Andrews, that the liquid and gaseous conditions of matter are only remote points in a continuous course of change. Nothing is at first sight more apparently distinct than the physical condition of water and aqueous vapour. At the boiling-point there is an entire breach of continuity, and the gas produced is subject to laws incomparably more simple than the liquid from which it arose. But Cagniard de la Tour showed that if we maintain a liquid under sufficient pressure its boiling point may be indefinitely raised, and yet the liquid will ultimately assume the gaseous condition with but a small increase of volume. Professor Andrews, recently following out this course of inquiry, has shown that liquid carbonic acid may, at a particular temperature (30°·92 C.), and under the pressure of 74 atmospheres, be at the same time in a state indistinguishable from that of liquid and gas. At higher pressures carbonic acid may be made to pass from a palpably liquid state to a truly gaseous state without any abrupt change whatever. As the pressure is greater the abruptness of the change from liquid to gas gradually decreases, and finally vanishes. Similar phenomena or an approximation to them have been observed in other liquids, and there is little doubt that we may make a wide generalisation, and assert that, under adequate pressure, every liquid might be made to pass into a gas without breach of continuity.[504] The liquid state, moreover, is considered by Professor Andrews to be but an intermediate step between the solid and gaseous conditions. There are various indications that the process of melting is not perfectly abrupt; and could experiments be made under adequate pressures, it is believed that every solid could be made to pass by insensible degrees into the state of liquid, and subsequently into that of gas.
[504] *Nature*, vol. ii. p. 278.
These discoveries appear to open the way to most important and fundamental generalisations, but it is probable that in many other cases phenomena now regarded as discrete may be shown to be different degrees of the same process. Graham was of opinion that chemical affinity differs but in degree from the ordinary attraction which holds different particles of a body together. He found that sulphuric acid continued to evolve heat when mixed even with the fiftieth equivalent of water, so that there seemed to be no distinct limit to chemical affinity. He concludes, “There is reason to believe that chemical affinity passes in its lowest degree into the attraction of aggregation.”[505]
[505] *Journal of the Chemical Society*, vol. viii. p. 51.
The atomic theory is well established, but its limits are not marked out. As Grove points out, we may by selecting sufficiently high multipliers express any combination or mixture of elements in terms of their equivalent weights.[506] Sir W. Thomson has suggested that the power which vegetable fibre, oatmeal, and other substances possess of attracting and condensing aqueous vapour is probably continuous, or, in fact, identical with capillary attraction, which is capable of interfering with the pressure of aqueous vapour and aiding its condensation.[507] There are many cases of so-called catalytic or surface action, such as the extraordinary power of animal charcoal for attracting organic matter, or of spongy platinum for condensing hydrogen, which can only be considered as exalted cases of a more general power of attraction. The number of substances which are decomposed by light in a striking manner is very limited; but many other substances, such as vegetable colours, are affected by long exposure; on the principle of continuity we might expect to find that all kinds of matter are more or less susceptible of change by the incidence of light rays.[508] It is the opinion of Grove that wherever an electric current passes there is a tendency to decomposition, a strain on the molecules, which when sufficiently intense leads to disruption. Even a metallic conducting wire may be regarded as tending to decomposition. Davy was probably correct in describing electricity as chemical affinity acting on masses, or rather, as Grove suggests, creating a disturbance through a chain of particles.[509] Laplace went so far as to suggest that all chemical phenomena may be results of the Newtonian law of attraction, applied to atoms of various mass and position; but the time is probably far distant when the progress of molecular philosophy and of mathematical methods will enable such a generalisation to be verified or refuted.
[506] *Correlation of Physical Forces*, 3rd edit. p. 184.
[507] *Philosophical Magazine*, 4th Series, vol. xlii. p. 451.
[508] Grove, *Correlation of Physical Forces*, 3rd edit. p. 118.
[509] Ibid. pp. 166, 199, &c.
*The Law of Continuity.*
Under the title of the Law of Continuity we may place many applications of the general principle of reasoning, that what is true of one case will be true of similar cases, and probably true of what are probably similar. Whenever we find that a law or similarity is rigorously fulfilled up to a certain point in time or space, we expect with a high degree of probability that it will continue to be fulfilled at least a little further. If we see part only of a circle, we naturally expect that the circular form will be continued in the part hidden from us. If a body has moved uniformly over a certain space, we expect that it will continue to move uniformly. The ground of such inferences is doubtless identical with that of other inductive inferences. In continuous motion every infinitely small space passed over constitutes a separate constituent fact, and had we perfect powers of observation the smallest finite motion would include an infinity of information, which, by the principles of the inverse method of probabilities, would enable us to infer with certainty to the next infinitely small portion of the path. But when we attempt to infer from one finite portion of a path to another finite portion, inference will be only more or less probable, according to the comparative lengths of the portions and the accuracy of observation; the longer our experience is, the more probable our inference will be; the greater the length of time or space over which the inference extends, the less probable.
This principle of continuity presents itself in nature in a great variety of forms and cases. It is familiarly expressed in the dictum *Natura non agit per saltum*. As Graham expressed the maxim, there are in nature no abrupt transitions, and the distinctions of class are never absolute.[510] There is always some notice--some forewarning of every phenomenon, and every change begins by insensible degrees, could we observe it with perfect accuracy. The cannon ball, indeed, is forced from the cannon in an inappreciable portion of time; the trigger is pulled, the fuze fired, the powder inflamed, the ball expelled, all simultaneously to our senses. But there is no doubt that time is occupied by every part of the process, and that the ball begins to move at first with infinite slowness. Captain Noble is able to measure by his chronoscope the progress of the shot in a 300-pounder gun, and finds that the whole motion within the barrel takes place in something less than one 200th part of a second. It is certain that no finite force can produce motion, except in a finite space of time. The amount of momentum communicated to a body is proportional to the accelerating force multiplied by the time during which it acts uniformly. Thus a slight force produces a great velocity only by long-continued action. In a powerful shock, like that of a railway collision, the stroke of a hammer on an anvil, or the discharge of a gun, the time is very short, and therefore the accelerating forces brought into play are exceedingly great, but never infinite. In the case of a large gun the powder in exploding is said to exert for a moment a force equivalent to at least 2,800,000 horses.
[510] *Philosophical Transactions*, 1861. *Chemical and Physical Researches*, p. 598.
Our belief in some of the fundamental laws of nature rests upon the principle of continuity. Galileo is held to be the first philosopher who consciously employed this principle in his arguments concerning the nature of motion, and it is certain that we can never by mere experience assure ourselves of the truth even of the first law of motion. *A material particle*, we are told, *when not acted on by extraneous forces will continue in the same state of rest or motion.* This may be true, but as we can find no body which is free from the action of extraneous causes, how are we to prove it? Only by observing that the less the amount of those forces the more nearly is the law found to be true. A ball rolled along rough ground is soon stopped; along a smooth pavement it continues longer in movement. A delicately suspended pendulum is almost free from friction against its supports, but it is gradually stopped by the resistance of the air; place it in the vacuous receiver of an air-pump and we find the motion much prolonged. A large planet like Jupiter experiences almost infinitely less friction, in comparison to its vast momentum, than we can produce experimentally, and we find in such a case that there is not the least evidence of the falsity of the law. Experience, then, informs us that we may approximate indefinitely to a uniform motion by sufficiently decreasing the disturbing forces. It is an act of inference which enables us to travel on beyond experience, and assert that, in the total absence of any extraneous force, motion would be absolutely uniform. The state of rest, again, is a limiting case in which motion is infinitely small or zero, to which we may attain, on the principle of continuity, by successively considering cases of slower and slower motion. There are many classes of phenomena, in which, by gradually passing from the apparent to the obscure, we can assure ourselves of the nature of phenomena which would otherwise be a matter of great doubt. Thus we can sufficiently prove in the manner of Galileo, that a musical sound consists of rapid uniform pulses, by causing strokes to be made at intervals which we gradually diminish until the separate strokes coalesce into a uniform hum or note. With great advantage we approach, as Tyndall says, the sonorous through the grossly mechanical. In listening to a great organ we cannot fail to perceive that the longest pipes, or their partial tones, produce a tremor and fluttering of the building. At the other extremity of the scale, there is no fixed limit to the acuteness of sounds which we can hear; some individuals can hear sounds too shrill for other ears, and as there is nothing in the nature of the atmosphere to prevent the existence of undulations far more rapid than any of which we are conscious, we may infer, by the principle of continuity, that such undulations probably exist.
There are many habitual actions which we perform we know not how. So rapidly are acts of minds accomplished that analysis seems impossible. We can only investigate them when in process of formation, observing that the best formed habit is slowly and continuously acquired, and it is in the early stages that we can perceive the rationale of the process.
Let it be observed that this principle of continuity must be held of much weight only in exact physical laws, those which doubtless repose ultimately upon the simple laws of motion. If we fearlessly apply the principle to all kinds of phenomena, we may often be right in our inferences, but also often wrong. Thus, before the development of spectrum analysis, astronomers had observed that the more they increased the powers of their telescopes the more nebulæ they could resolve into distinct stars. This result had been so often found true that they almost irresistibly assumed that all nebulæ would be ultimately resolved by telescopes of sufficient power; yet Huggins has in recent years proved by the spectroscope, that certain nebulæ are actually gaseous, and in a truly nebulous state.
The principle of continuity must have been continually employed in the inquiries of Galileo, Newton, and other experimental philosophers, but it appears to have been distinctly formulated for the first time by Leibnitz. He at least claims to have first spoken of “the law of continuity” in a letter to Bayle, printed in the *Nouvelles de la République des Lettres*, an extract from which is given in Erdmann’s edition of Leibnitz’s works, p. 104, under the title “Sur un Principe Général utile à l’explication des Lois de la Nature.”[511] It has indeed been asserted that the doctrine of the *latens processus* of Francis Bacon involves the principle of continuity,[512] but I think that this doctrine, like that of the *natures* of substances, is merely a vague statement of the principle of causation.
[511] *Life of Sir W. Hamilton*, p. 439.
[512] Powell’s *History of Natural Philosophy*, p. 201. *Novum Organum*, bk. ii. Aphorisms 5–7.
*Failure of the Law of Continuity.*
There are certain cautions which must be given as to the application of the principle of continuity. In the first place, where this principle really holds true, it may seem to fail owing to our imperfect means of observation. Though a physical law may not admit of perfectly abrupt change, there is no limit to the approach which it may make to abruptness. When we warm a piece of very cold ice, the absorption of heat, the temperature, and the dilatation of the ice vary according to apparently simple laws until we come to the zero of the Centigrade scale. Everything is then changed; an enormous absorption of heat takes place without any rise of temperature, and the volume of the ice decreases as it changes into water. Unless carefully investigated, this change appears to be perfectly abrupt; but accurate observation seems to show that there is a certain forewarning; the ice does not turn into water all at once, but through a small fraction of a degree the change is gradual. All the phenomena concerned, if measured very exactly, would be represented not by angular lines, but continuous curves, undergoing rapid flexures; and we may probably assert with safety that between whatever points of temperature we examine ice, there would be found some indication, though almost infinitesimally small, of the apparently abrupt change which was to occur at a higher temperature. It might also be pointed out that the important and apparently simple physical laws, such as those of Boyle and Mariotte, Dalton and Gay-Lussac, &c., are only approximately true, and the divergences from the simple laws are forewarnings of abrupt changes, which would otherwise break the law of continuity.
Secondly, it must be remembered that mathematical laws of some complexity will probably present singular cases or negative results, which may bear the appearance of discontinuity, as when the law of retraction suddenly yields us with perfect abruptness the phenomenon of total internal reflection. In the undulatory theory, however, there is no real change of law between refraction and reflection. Faraday in the earlier part of his career found so many substances possessing magnetic power, that he ventured on a great generalisation, and asserted that all bodies shared in the magnetic property of iron. His mistake, as he afterwards discovered, consisted in overlooking the fact that though magnetic in a certain sense, some substances have negative magnetism, and are repelled instead of being attracted by the magnet.
Thirdly, where we might expect to find a uniform mathematical law prevailing, the law may undergo abrupt change at singular points, and actual discontinuity may arise. We may sometimes be in danger of treating under one law phenomena which really belong to different laws. For instance, a spherical shell of uniform matter attracts an external particle of matter with a force varying inversely as the square of the distance from the centre of the sphere. But this law only holds true so long as the particle is external to the shell. Within the shell the law is wholly different, and the aggregate gravity of the sphere becomes zero, the force in every direction being neutralised by an exactly equal opposite force. If an infinitely small particle be in the superficies of a sphere, the law is again different, and the attractive power of the shell is half what it would be with regard to particles infinitely close to the surface of the shell. Thus in approaching the centre of a shell from a distance, the force of gravity shows double discontinuity in passing through the shell.[513]
[513] Thomson and Tait, *Treatise on Natural Philosophy*, vol. i. pp. 346–351.
It may admit of question, too, whether discontinuity is really unknown in nature. We perpetually do meet with events which are real breaks upon the previous law, though the discontinuity may be a sign that some independent cause has come into operation. If the ordinary course of the tides is interrupted by an enormous irregular wave, we attribute it to an earthquake, or some gigantic natural disturbance. If a meteoric stone falls upon a person and kills him, it is clearly a discontinuity in his life, of which he could have had no anticipation. A sudden sound may pass through the air neither preceded nor followed by any continuous effect. Although, then, we may regard the Law of Continuity as a principle of nature holding rigorously true in many of the relations of natural forces, it seems to be a matter of difficulty to assign the limits within which the law is verified. Much caution is required in its application.
*Negative Arguments on the Principle of Continuity.*
Upon the principle of continuity we may sometimes found arguments of great force which prove an hypothesis to be impossible, because it would involve a continual repetition of a process *ad infinitum*, or else a purely arbitrary breach at some point. Bonnet’s famous theory of reproduction represented every living creature as containing germs which were perfect representatives of the next generation, so that on the same principle they necessarily included germs of the next generation, and so on indefinitely. The theory was sufficiently refuted when once clearly stated, as in the following poem called the Universe,[514] by Henry Baker:--
“Each seed includes a plant: that plant, again, Has other seeds, which other plants contain: Those other plants have all their seeds, and those More plants again, successively inclose.
“Thus, ev’ry single berry that we find, Has, really, in itself whole forests of its kind, Empire and wealth one acorn may dispense, By fleets to sail a thousand ages hence.”
[514] *Philosophical Transactions* (1740), vol. xli. p. 454.
The general principle of inference, that what we know of one case must be true of similar cases, so far as they are similar, prevents our asserting anything which we cannot apply time after time under the same circumstances. On this principle Stevinus beautifully demonstrated that weights resting on two inclined planes and balancing each other must be proportional to the lengths of the planes between their apex and a horizontal plane. He imagined a uniform endless chain to be hung over the planes, and to hang below in a symmetrical festoon. If the chain were ever to move by gravity, there would be the same reason for its moving on for ever, and thus producing a perpetual motion. As this is absurd, the portions of the chain lying on the planes, and equal in length to the planes, must balance each other. On similar grounds we may disprove the existence of any *self-moving machine*; for if it could once alter its own state of motion or rest, in however small a degree, there is no reason why it should not do the like time after time *ad infinitum*. Newton’s proof of his third law of motion, in the case of gravity, is of this character. For he remarks that if two gravitating bodies do not exert exactly equal forces in opposite directions, the one exerting the strongest pull will carry both away, and the two bodies will move off into space together with velocity increasing *ad infinitum*. But though the argument might seem sufficiently convincing, Newton in his characteristic way made an experiment with a loadstone and iron floated upon the surface of water.[515] In recent years the very foundation of the principle of conservation of energy has been placed on the assumption that it is impossible by any combination of natural bodies to produce force continually from nothing.[516] The principle admits of application in various subtle forms.
[515] *Principia*, bk. i. Law iii. Corollary 6.
[516] Helmholtz, Taylor’s *Scientific Memoirs* (1853), vol. vi. p. 118.
Lucretius attempted to prove, by a most ingenious argument of this kind, that matter must be indestructible. For if a finite quantity, however small, were to fall out of existence in any finite time, an equal quantity might be supposed to lapse in every equal interval of time, so that in the infinity of past time the universe must have ceased to exist.[517] But the argument, however ingenious, seems to fail at several points. If past time be infinite, why may not matter have been created infinite also? It would be most reasonable, again, to suppose the matter destroyed in any time to be proportional to the matter then remaining, and not to the original quantity; under this hypothesis even a finite quantity of original matter could never wholly disappear from the universe. For like reasons we cannot hold that the doctrine of the conservation of energy is really proved, or can ever be proved to be absolutely true, however probable it may be regarded.
[517] *Lucretius*, bk. i. lines 232–264.
*Tendency to Hasty Generalisation.*
In spite of all the powers and advantages of generalisation, men require no incitement to generalise; they are too apt to draw hasty and ill-considered inferences. As Francis Bacon said, our intellects want not wings, but rather weights of lead to moderate their course.[518] The process is inevitable to the human mind; it begins with childhood and lasts through the second childhood. The child that has once been hurt fears the like result on all similar occasions, and can with difficulty be made to distinguish between case and case. It is caution and discrimination in the adoption of conclusions that we have chiefly to learn, and the whole experience of life is one continued lesson to this effect. Baden Powell has excellently described this strong natural propensity to hasty inference, and the fondness of the human mind for tracing resemblances real or fanciful. “Our first inductions,” he says,[519] “are always imperfect and inconclusive; we advance towards real evidence by successive approximations; and accordingly we find false generalisation the besetting error of most first attempts at scientific research. The faculty to generalise accurately and philosophically requires large caution and long training, and is not fully attained, especially in reference to more general views, even by some who may properly claim the title of very accurate scientific observers in a more limited field. It is an intellectual habit which acquires immense and accumulating force from the contemplation of wider analogies.”
[518] *Novum Organum*, bk. 1 Aphorism 104.
[519] *The Unity of Worlds and of Nature*, 2nd edit. p. 116.
Hasty and superficial generalisations have always been the bane of science, and there would be no difficulty in finding endless illustrations. Between things which are the same in number there is a certain resemblance, namely in number; but in the infancy of science men could not be persuaded that there was not a deeper resemblance implied in that of number. Pythagoras was not the inventor of a mystical science of number. In the ancient Oriental religions the seven metals were connected with the seven planets, and in the seven days of the week we still have, and probably always shall have, a relic of the septiform system ascribed by Dio Cassius to the ancient Egyptians. The disciples of Pythagoras carried the doctrine of the number seven into great detail. Seven days are mentioned in Genesis; infants acquire their teeth at the end of seven months; they change them at the end of seven years; seven feet was the limit of man’s height; every seventh year was a climacteric or critical year, at which a change of disposition took place. Then again there were the seven sages of Greece, the seven wonders of the world, the seven rites of the Grecian games, the seven gates of Thebes, and the seven generals destined to conquer that city.
In natural science there were not only the seven planets, and the seven metals, but also the seven primitive colours, and the seven tones of music. So deep a hold did this doctrine take that we still have its results in many customs, not only in the seven days of the week, but the seven years’ apprenticeship, puberty at fourteen years, the second climacteric, and legal majority at twenty-one years, the third climacteric. The idea was reproduced in the seven sacraments of the Roman Catholic Church, and the seven year periods of Comte’s grotesque system of domestic worship. Even in scientific matters the loftiest intellects have occasionally yielded, as when Newton was misled by the analogy between the seven tones of music and the seven colours of his spectrum. Other numerical analogies, though rejected by Galileo, held Kepler in thraldom; no small part of Kepler’s labours during seventeen years was spent upon numerical and geometrical analogies of the most baseless character; and he gravely held that there could not be more than six planets, because there were not more than five regular solids. Even the genius of Huyghens did not prevent him from inferring that but one satellite could belong to Saturn, because, with those of Jupiter and the Earth, it completed the perfect number of six. A whole series of other superstitions and fallacies attach to the numbers six and nine.
It is by false generalisation, again, that the laws of nature have been supposed to possess that perfection which we attribute to simple forms and relations. The heavenly bodies, it was held, must move in circles, for the circle was the perfect figure. Newton seemed to adopt the questionable axiom that nature always proceeds in the simplest way; in stating his first rule of philosophising, he adds:[520] “To this purpose the philosophers say, that nature does nothing in vain, when less will serve; for nature is pleased with simplicity, and affects not the pomp of superfluous causes.” Keill lays down[521] as an axiom that “The causes of natural things are such, as are the most simple, and are sufficient to explain the phenomena: for nature always proceeds in the simplest and most expeditious method; because by this manner of operating the Divine Wisdom displays itself the more.” If this axiom had any clear grounds of truth, it would not apply to proximate laws; for even when the ultimate law is simple the results may be infinitely diverse, as in the various elliptic, hyperbolic, parabolic, or circular orbits of the heavenly bodies. Simplicity is naturally agreeable to a mind of limited powers, but to an infinite mind all things are simple.
[520] *Principia*, bk. iii, *ad initium*.
[521] Keill, *Introduction to Natural Philosophy*, p. 89.
Every great advance in science consists in a great generalisation, pointing out deep and subtle resemblances. The Copernican system was a generalisation, in that it classed the earth among the planets; it was, as Bishop Wilkins expressed it, “the discovery of a new planet,” but it was opposed by a more shallow generalisation. Those who argued from the condition of things upon the earth’s surface, thought that every object must be attached to and rest upon something else. Shall the earth, they said, alone be free? Accustomed to certain special results of gravity they could not conceive its action under widely different circumstances.[522] No hasty thinker could seize the deep analogy pointed out by Horrocks between a pendulum and a planet, true in substance though mistaken in some details. All the advances of modern science rise from the conception of Galileo, that in the heavenly bodies, however apparently different their condition, we shall ultimately recognise the same fundamental principles of mechanical science which are true on earth.
[522] Jeremiæ Horroccii *Opera Posthuma* (1673), pp. 26, 27.Return to passage
Generalisation is the great prerogative of the intellect, but it is a power only to be exercised safely with much caution and after long training. Every mind must generalise, but there are the widest differences in the depth of the resemblances discovered and the care with which the discovery is verified. There seems to be an innate power of insight which a few men have possessed pre-eminently, and which enabled them, with no exemption indeed from labour or temporary error, to discover the one in the many. Minds of excessive acuteness may exist, which have yet only the powers of minute discrimination, and of storing up, in the treasure-house of memory, vast accumulations of words and incidents. But the power of discovery belongs to a more restricted class of minds. Laplace said that, of all inventors who had contributed the most to the advancement of human knowledge, Newton and Lagrange appeared to possess in the highest degree the happy tact of distinguishing general principles among a multitude of objects enveloping them, and this tact he conceived to be the true characteristic of scientific genius.[523]
[523] Young’s *Works*, vol. ii. p. 564.Return to passage