XX. METHOD OF VARIATIONS
Experiments may be of two kinds, experiments of simple fact, and experiments of quantity. In the first class of experiments we combine certain conditions, and wish to ascertain whether or not a certain effect of any quantity exists. Hooke wished to ascertain whether or not there was any difference in the force of gravity at the top and bottom of St. Paul’s Cathedral. The chemist continually performs analyses for the purpose of ascertaining whether or not a given element exists in a particular mineral or mixture; all such experiments and analyses are qualitative rather than quantitative, because though the result may be more or less, the particular amount of the result is not the object of the inquiry.
So soon, however, as a result is known to be discoverable, the scientific man ought to proceed to the quantitative inquiry, how great a result follows from a certain amount of the conditions which are supposed to constitute the cause? The possible numbers of experiments are now infinitely great, for every variation in a quantitative condition will usually produce a variation in the amount of the effect. The method of variation which thus arises is no narrow or special method, but it is the general application of experiment to phenomena capable of continuous variation. As Mr. Fowler has well remarked,[356] the observation of variations is really an integration of a supposed infinite number of applications of the so-called method of difference, that is of experiment in its perfect form.
[356] *Elements of Inductive Logic*, 1st edit. p. 175.
In induction we aim at establishing a general law, and if we deal with quantities that law must really be expressed more or less obviously in the form of an equation, or equations. We treat as before of conditions, and of what happens under those conditions. But the conditions will now vary, not in quality, but quantity, and the effect will also vary in quantity, so that the result of quantitative induction is always to arrive at some mathematical expression involving the quantity of each condition, and expressing the quantity of the result. In other words, we wish to know what function the effect is of its conditions. We shall find that it is one thing to obtain the numerical results, and quite another thing to detect the law obeyed by those results, the latter being an operation of an inverse and tentative character.
*The Variable and the Variant.*
Almost every series of quantitative experiments is directed to obtain the relation between the different values of one quantity which is varied at will, and another quantity which is caused thereby to vary. We may conveniently distinguish these as respectively the *variable* and the *variant*. When we are examining the effect of heat in expanding bodies, heat, or one of its dimensions, temperature, is the variable, length the variant. If we compress a body to observe how much it is thereby heated, pressure, or it may be the dimensions of the body, forms the variable, heat the variant. In the thermo-electric pile we make heat the variable and measure electricity as the variant. That one of the two measured quantities which is an antecedent condition of the other will be the variable.
It is always convenient to have the variable entirely under our command. Experiments may indeed be made with accuracy, provided we can exactly measure the variable at the moment when the quantity of the effect is determined. But if we have to trust to the action of some capricious force, there may be great difficulty in making exact measurements, and those results may not be disposed over the whole range of quantity in a convenient manner. It is one prime object of the experimenter, therefore, to obtain a regular and governable supply of the force which he is investigating. To determine correctly the efficiency of windmills, when the natural winds were constantly varying in force, would be exceedingly difficult. Smeaton, therefore, in his experiments on the subject, created a uniform wind of the required force by moving his models against the air on the extremity of a revolving arm.[357] The velocity of the wind could thus be rendered greater or less, it could be maintained uniform for any length of time, and its amount could be exactly ascertained. In determining the laws of the chemical action of light it would be out of the question to employ the rays of the sun, which vary in intensity with the clearness of the atmosphere, and with every passing cloud. One great difficulty in photometry and the investigation of the chemical action of light consists in obtaining a uniform and governable source of light rays.[358]
[357] *Philosophical Transactions*, vol. li. p. 138; abridgment, vol. xi. p. 355.
[358] See Bunsen and Roscoe’s researches, in *Philosophical Transactions* (1859), vol. cxlix. p. 880, &c., where they describe a constant flame of carbon monoxide gas.
Fizeau’s method of measuring the velocity of light enabled him to appreciate the time occupied by light in travelling through a distance of eight or nine thousand metres. But the revolving mirror of Wheatstone subsequently enabled Foucault and Fizeau to measure the velocity in a space of four metres. In this latter method there was the advantage that various media could be substituted for air, and the temperature, density, and other conditions of the experiment could be accurately governed and measured.
*Measurement of the Variable.*
There is little use in obtaining exact measurements of an effect unless we can also exactly measure its conditions.
It is absurd to measure the electrical resistance of a piece of metal, its elasticity, tenacity, density, or other physical qualities, if these vary, not only with the minute impurities of the metal, but also with its physical condition. If the same bar changes its properties by being heated and cooled, and we cannot exactly define the state in which it is at any moment, our care in measuring will be wasted, because it can lead to no law. It is of little use to determine very exactly the electric conductibility of carbon, which as graphite or gas carbon conducts like a metal, as diamond is almost a non-conductor, and in several other forms possesses variable and intermediate powers of conduction. It will be of use only for immediate practical applications. Before measuring these we ought to have something to measure of which the conditions are capable of exact definition, and to which at a future time we can recur. Similarly the accuracy of our measurement need not much surpass the accuracy with which we can define the conditions of the object treated.
The speed of electricity in passing through a conductor mainly depends upon the inductive capacity of the surrounding substances, and, except for technical or special purposes, there is little use in measuring velocities which in some cases are one hundred times as great as in other cases. But the maximum speed of electric conduction is probably a constant quantity of great scientific importance, and according to Prof. Clerk Maxwell’s determination in 1868 is 174,800 miles per second, or little less than that of light. The true boiling point of water is a point on which practical thermometry depends, and it is highly important to determine that point in relation to the absolute thermometric scale. But when water free from air and impurity is heated there seems to be no definite limit to the temperature it may reach, a temperature of 180° Cent. having been actually observed. Such temperatures, therefore, do not require accurate measurement. All meteorological measurements depending on the accidental condition of the sky are of far less importance than physical measurements in which such accidental conditions do not intervene. Many profound investigations depend upon our knowledge of the radiant energy continually poured upon the earth by the sun; but this must be measured when the sky is perfectly clear, and the absorption of the atmosphere at its minimum. The slightest interference of cloud destroys the value of such a measurement, except for meteorological purposes, which are of vastly less generality and importance. It is seldom useful, again, to measure the height of a snow-covered mountain within a foot, when the thickness of the snow alone may cause it to vary 25 feet or more, when in short the height itself is indefinite to that extent.[359]
[359] Humboldt’s *Cosmos* (Bohn), vol. i. p. 7.
*Maintenance of Similar Conditions.*
Our ultimate object in induction must be to obtain the complete relation between the conditions and the effect, but this relation will generally be so complex that we can only attack it in detail. We must, as far as possible, confine the variation to one condition at a time, and establish a separate relation between each condition and the effect. This is at any rate the first step in approximating to the complete law, and it will be a subsequent question how far the simultaneous variation of several conditions modifies their separate actions. In many experiments, indeed, it is only one condition which we wish to study, and the others are interfering forces which we would avoid if possible. One of the conditions of the motion of a pendulum is the resistance of the air, or other medium in which it swings; but when Newton was desirous of proving the equal gravitation of all substances, he had no interest in the air. His object was to observe a single force only, and so it is in a great many other experiments. Accordingly, one of the most important precautions in investigation consists in maintaining all conditions constant except that which is to be studied. As that admirable experimental philosopher, Gilbert, expressed it,[360] “There is always need of similar preparation, of similar figure, and of equal magnitude, for in dissimilar and unequal circumstances the experiment is doubtful.”
[360] Gilbert, *De Magnete*, p. 109.
In Newton’s decisive experiment similar conditions were provided for, with the simplicity which characterises the highest art. The pendulums of which the oscillations were compared consisted of equal boxes of wood, hanging by equal threads, and filled with different substances, so that the total weights should be equal and the centres of oscillation at the same distance from the points of suspension. Hence the resistance of the air became approximately a matter of indifference; for the outward size and shape of the pendulums being the same, the absolute force of resistance would be the same, so long as the pendulums vibrated with equal velocity; and the weights being equal the resistance would diminish the velocity equally. Hence if any inequality were observed in the vibrations of the two pendulums, it must arise from the only circumstance which was different, namely the chemical nature of the matter within the boxes. No inequality being observed, the chemical nature of substances can have no appreciable influence upon the force of gravitation.[361]
[361] *Principia*, bk. iii. Prop. vi.
A beautiful experiment was devised by Dr. Joule for the purpose of showing that the gain or loss of heat by a gas is connected, not with the mere change of its volume and density, but with the energy received or given out by the gas. Two strong vessels, connected by a tube and stopcock, were placed in water after the air had been exhausted from one vessel and condensed in the other to the extent of twenty atmospheres. The whole apparatus having been brought to a uniform temperature by agitating the water, and the temperature having been exactly observed, the stopcock was opened, so that the air at once expanded and filled the two vessels uniformly. The temperature of the water being again noted was found to be almost unchanged. The experiment was then repeated in an exactly similar manner, except that the strong vessels were placed in separate portions of the water. Now cold was produced in the vessel from which the air rushed, and an almost exactly equal quantity of heat appeared in that to which it was conducted. Thus Dr. Joule clearly proved that rarefaction produces as much heat as cold, and that only when there is disappearance of mechanical energy will there be production of heat.[362] What we have to notice, however, is not so much the result of the experiment, as the simple manner in which a single change in the apparatus, the separation of the portions of water surrounding the air vessels, is made to give indications of the utmost significance.
[362] *Philosophical Magazine*, 3rd Series, vol. xxvi. p. 375.
*Collective Experiments.*
There is an interesting class of experiments which enable us to observe a number of quantitative results in one act. Generally speaking, each experiment yields us but one number, and before we can approach the real processes of reasoning we must laboriously repeat measurement after measurement, until we can lay out a curve of the variation of one quantity as depending on another. We can sometimes abbreviate this labour, by making a quantity vary in different parts of the same apparatus through every required amount. In observing the height to which water rises by the capillary attraction of a glass vessel, we may take a series of glass tubes of different bore, and measure the height through which it rises in each. But if we take two glass plates, and place them vertically in water, so as to be in contact at one vertical side, and slightly separated at the other side, the interval between the plates varies through every intermediate width, and the water rises to a corresponding height, producing at its upper surface a hyperbolic curve.
The absorption of light in passing through a coloured liquid may be beautifully shown by enclosing the liquid in a wedge-shaped glass, so that we have at a single glance an infinite variety of thicknesses in view. As Newton himself remarked, a red liquid viewed in this manner is found to have a pale yellow colour at the thinnest part, and it passes through orange into red, which gradually becomes of a deeper and darker tint.[363] The effect may be noticed in a conical wine-glass. The prismatic analysis of light from such a wedge-shaped vessel discloses the reason, by exhibiting the progressive absorption of different rays of the spectrum as investigated by Dr. J. H. Gladstone.[364]
[363] *Opticks*, 3rd edit. p. 159.
[364] Watts, *Dictionary of Chemistry*, vol. iii. p. 637.
A moving body may sometimes be made to mark out its own course, like a shooting star which leaves a tail behind it. Thus an inclined jet of water exhibits in the clearest manner the parabolic path of a projectile. In Wheatstone’s Kaleidophone the curves produced by the combination of vibrations of different ratios are shown by placing bright reflective buttons on the tops of wires of various forms. The motions are performed so quickly that the eye receives the impression of the path as a complete whole, just as a burning stick whirled round produces a continuous circle. The laws of electric induction are beautifully shown when iron filings are brought under the influence of a magnet, and fall into curves corresponding to what Faraday called the Lines of Magnetic Force. When Faraday tried to define what he meant by his lines of force, he was obliged to refer to the filings. “By magnetic curves,” he says,[365] “I mean lines of magnetic forces which would be depicted by iron filings.” Robison had previously produced similar curves by the action of frictional electricity, and from a mathematical investigation of the forms of such curves we may infer that magnetic and electric attractions obey the general law of emanation, that of the inverse square of the distance. In the electric brush we have a similar exhibition of the laws of electric attraction.
[365] *Faraday’s Life*, by Bence Jones, vol. ii. p. 5.
There are several branches of science in which collective experiments have been used with great advantage. Lichtenberg’s electric figures, produced by scattering electrified powder on an electrified resin cake, so as to show the condition of the latter, suggested to Chladni the notion of discovering the state of vibration of plates by strewing sand upon them. The sand collects at the points where the motion is least, and we gain at a glance a comprehension of the undulations of the plate. To this method of experiment we owe the beautiful observations of Savart. The exquisite coloured figures exhibited by plates of crystal, when examined by polarised light, afford a more complicated example of the same kind of investigation. They led Brewster and Fresnel to an explanation of the properties of the optic axes of crystals. The unequal conduction of heat in crystalline substances has also been shown in a similar manner, by spreading a thin layer of wax over the plate of crystal, and applying heat to a single point. The wax then melts in a circular or elliptic area according as the rate of conduction is uniform or not. Nor should we forget that Newton’s rings were an early and most important instance of investigations of the same kind, showing the effects of interference of light undulations of all magnitudes at a single view. Herschel gave to all such opportunities of observing directly the results of a general law, the name of *Collective Instances*,[366] and I propose to adopt the name *Collective Experiments*.
[366] *Preliminary Discourse*, &c., p. 185.
Such experiments will in many subjects only give the first hint of the nature of the law in question, but will not admit of any exact measurements. The parabolic form of a jet of water may well have suggested to Galileo his views concerning the path of a projectile; but it would not serve now for the exact investigation of the laws of gravity. It is unlikely that capillary attraction could be exactly measured by the use of inclined plates of glass, and tubes would probably be better for precise investigation. As a general rule, these collective experiments would be most useful for popular illustration. But when the curves are of a precise and permanent character, as in the coloured figures produced by crystalline plates, they may admit of exact measurement. Newton’s rings and diffraction fringes allow of very accurate measurements.
Under collective experiments we may perhaps place those in which we render visible the motions of gas or liquid by diffusing some opaque substance in it. The behaviour of a body of air may often be studied in a beautiful way by the use of smoke, as in the production of smoke rings and jets. In the case of liquids lycopodium powder is sometimes employed. To detect the mixture of currents or strata of liquid, I employed very dilute solutions of common salt and silver nitrate, which produce a visible cloud wherever they come into contact.[367] Atmospheric clouds often reveal to us the movements of great volumes of air which would otherwise be quite unapparent.
[367] *Philosophical Magazine*, July, 1857, 4th Series, vol. xiv. p. 24.
*Periodic Variations.*
A large class of investigations is concerned with Periodic Variations. We may define a periodic phenomenon as one which, with the uniform change of the variable, returns time after time to the same value. If we strike a pendulum it presently returns to the point from which we disturbed it, and while time, the variable, progresses uniformly, it goes on making excursions and returning, until stopped by the dissipation of its energy. If one body in space approaches by gravity towards another, they will revolve round each other in elliptic orbits, and return for an indefinite number of times to the same relative positions. On the other hand a single body projected into empty space, free from the action of any extraneous force, would go on moving for ever in a straight line, according to the first law of motion. In the latter case the variation is called *secular*, because it proceeds during ages in a similar manner, and suffers no περίοδος or going round. It may be doubted whether there really is any motion in the universe which is not periodic. Mr. Herbert Spencer long since adopted the doctrine that all motion is ultimately rhythmical,[368] and abundance of evidence may be adduced in favour of his view.
[368] *First Principles*, 3rd edit. chap. x. p. 253.
The so-called secular acceleration of the moon’s motion is certainly periodic, and as, so far as we can tell, no body is beyond the attractive power of other bodies, rectilinear motion becomes purely hypothetical, or at least infinitely improbable. All the motions of all the stars must tend to become periodic. Though certain disturbances in the planetary system seem to be uniformly progressive, Laplace is considered to have proved that they really have their limits, so that after an immense time, all the planetary bodies might return to the same places, and the stability of the system be established. Such a theory of periodic stability is really hypothetical, and does not take into account phenomena resulting in the dissipation of energy, which may be a really secular process. For our present purposes we need not attempt to form an opinion on such questions. Any change which does not present the appearance of a periodic character will be empirically regarded as a secular change, so that there will be plenty of non-periodic variations.
The variations which we produce experimentally will often be non-periodic. When we communicate heat to a gas it increases in bulk or pressure, and as far as we can go the higher the temperature the higher the pressure. Our experiments are of course restricted in temperature both above and below, but there is every reason to believe that the bulk being the same, the pressure would never return to the same point at any two different temperatures. We may of course repeatedly raise and lower the temperature at regular or irregular intervals entirely at our will, and the pressure of the gas will vary in like manner and exactly at the same intervals, but such an arbitrary series of changes would not constitute Periodic Variation. It would constitute a succession of distinct experiments, which would place beyond reasonable doubt the connexion of cause and effect.
Whenever a phenomenon recurs at equal or nearly equal intervals, there is, according to the theory of probability, considerable evidence of connexion, because if the recurrences were entirely casual it is unlikely that they would happen at equal intervals. The fact that a brilliant comet had appeared in the years 1301, 1378, 1456, 1531, 1607, and 1682 gave considerable presumption in favour of the identity of the body, apart from similarity of the orbit. There is nothing which so fascinates the attention of men as the recurrence time after time of some unusual event. Things and appearances which remain ever the same, like mountains and valleys, fail to excite the curiosity of a primitive people. It has been remarked by Laplace that even in his day the rising of Venus in its brightest phase never failed to excite surprise and interest. So there is little doubt that the first germ of science arose in the attention given by Eastern people to the changes of the moon and the motions of the planets. Perhaps the earliest astronomical discovery consisted in proving the identity of the morning and evening stars, on the grounds of their similarity of aspect and invariable alternation.[369] Periodical changes of a somewhat complicated kind must have been understood by the Chaldeans, because they were aware of the cycle of 6585 days or 19 years which brings round the new and full moon upon the same days, hours, and even minutes of the year. The earliest efforts of scientific prophecy were founded upon this knowledge, and if at present we cannot help wondering at the precise anticipations of the nautical almanack, we may imagine the wonder excited by such predictions in early times.
[369] Laplace, *System of the World*, vol. i. pp. 50, 54, &c.
*Combined Periodic Changes.*
We shall seldom find a body subject to a single periodic variation, and free from other disturbances. We may expect the periodic variation itself to undergo variation, which may possibly be secular, but is more likely to prove periodic; nor is there any limit to the complication of periods beyond periods, or periods within periods, which may ultimately be disclosed. In studying a phenomenon of rhythmical character we have a succession of questions to ask. Is the periodic variation uniform? If not, is the change uniform? If not, is the change itself periodic? Is that new period uniform, or subject to any other change, or not? and so on *ad infinitum*.
In some cases there may be many distinct causes of periodic variations, and according to the principle of the superposition of small effects, to be afterwards considered, these periodic effects will be simply added together, or at least approximately so, and the joint result may present a very complicated subject of investigation. The tides of the ocean consist of a series of superimposed undulations. Not only are there the ordinary semi-diurnal tides caused by sun and moon, but a series of minor tides, such as the lunar diurnal, the solar diurnal, the lunar monthly, the lunar fortnightly, the solar annual and solar semi-annual are gradually being disentangled by the labours of Sir W. Thomson, Professor Haughton and others.
Variable stars present interesting periodic phenomena; while some stars, δ Cephei for instance, are subject to very regular variations, others, like Mira Ceti, are less constant in the degrees of brilliancy which they attain or the rapidity of the changes, possibly on account of some longer periodic variation.[370] The star β Lyræ presents a double maximum and minimum in each of its periods of nearly 13 days, and since the discovery of this variation the period in a period has probably been on the increase. “At first the variability was more rapid, then it became gradually slower; and this decrease in the length of time reached its limit between the years 1840 and 1844. During that time its period was nearly invariable; at present it is again decidedly on the decrease.”[371] The tracing out of such complicated variations presents an unlimited field for interesting investigation. The number of such variable stars already known is considerable, and there is no reason to suppose that any appreciable fraction of the whole number has yet been detected.
[370] Herschel’s *Outlines of Astronomy*, 4th edit. pp. 555–557.
[371] Humboldt’s *Cosmos* (Bohn), vol. iii. p. 229.
*Principle of Forced Vibrations.*
Investigations of the connection of periodic causes and effects rest upon a principle, which has been demonstrated by Sir John Herschel for some special cases, and clearly explained by him in several of his works.[372] The principle may be formally stated in the following manner: “If one part of any system connected together either by material ties, or by the mutual attractions of its members, be continually maintained by any cause, whether inherent in the constitution of the system or external to it, in a state of regular periodic motion, that motion will be propagated throughout the whole system, and will give rise, in every member of it, and in every part of each member, to periodic movements executed in equal periods, with that to which they owe their origin, though not necessarily synchronous with them in their maxima and minima.” The meaning of the proposition is that the effect of a periodic cause will be periodic, and will recur at intervals equal to those of the cause. Accordingly when we find two phenomena which do proceed, time after time, through changes of the same period, there is much probability that they are connected. In this manner, doubtless, Pliny correctly inferred that the cause of the tides lies in the sun and the moon, the intervals between successive high tides being equal to the intervals between the moon’s passage across the meridian. Kepler and Descartes too admitted the connection previous to Newton’s demonstration of its precise nature. When Bradley discovered the apparent motion of the stars arising from the aberration of light, he was soon able to attribute it to the earth’s annual motion, because it went through its phases in a year.
[372] *Encyclopædia Metropolitana*, art. *Sound*, § 323; *Outlines of Astronomy*, 4th edit., § 650. pp. 410, 487–88; *Meteorology, Encyclopædia Britannica*, Reprint, p. 197.
The most beautiful instance of induction concerning periodic changes which can be cited, is the discovery of an eleven-year period in various meteorological phenomena. It would be difficult to mention any two things apparently more disconnected than the spots upon the sun and auroras. As long ago as 1826, Schwabe commenced a regular series of observations of the spots upon the sun, which has been continued to the present time, and he was able to show that at intervals of about eleven years the spots increased much in size and number. Hardly was this discovery made known, when Lamont pointed out a nearly equal period of variation in the declination of the magnetic needle. Magnetic storms or sudden disturbances of the needle were next shown to take place most frequently at the times when sun-spots were prevalent, and as auroras are generally coincident with magnetic storms, these phenomena were brought into the cycle. It has since been shown by Professor Piazzi Smyth and Mr. E. J. Stone, that the temperature of the earth’s surface as indicated by sunken thermometers gives some evidence of a like period. The existence of a periodic cause having once been established, it is quite to be expected, according to the principle of forced vibrations, that its influence will be detected in all meteorological phenomena.
*Integrated Variations.*
In considering the various modes in which one effect may depend upon another, we must set in a distinct class those which arise from the accumulated effects of a constantly acting cause. When water runs out of a cistern, the velocity of motion depends, according to Torricelli’s theorem, on the height of the surface of the water above the vent; but the amount of water which leaves the cistern in a given time depends upon the aggregate result of that velocity, and is only to be ascertained by the mathematical process of integration. When one gravitating body falls towards another, the force of gravity varies according to the inverse square of the distance; to obtain the velocity produced we must integrate or sum the effects of that law; and to obtain the space passed over by the body in a given time, we must integrate again.
In periodic variations the same distinction must be drawn. The heating power of the sun’s rays at any place on the earth varies every day with the height attained, and is greatest about noon; but the temperature of the air will not be greatest at the same time. This temperature is an integrated effect of the sun’s heating power, and as long as the sun is able to give more heat to the air than the air loses in other ways, the temperature continues to rise, so that the maximum is deferred until about 3 P.M. Similarly the hottest day of the year falls, on an average, about one month later than the summer solstice, and all the seasons lag about a month behind the motions of the sun. In the case of the tides, too, the effect of the moon’s attractive power is never greatest when the power is greatest; the effect always lags more or less behind the cause. Yet the intervals between successive tides are equal, in the absence of disturbance, to the intervals between the passages of the moon across the meridian. Thus the principle of forced vibrations holds true.
In periodic phenomena, however, curious results sometimes follow from the integration of effects. If we strike a pendulum, and then repeat the stroke time after time at the same part of the vibration, all the strokes concur in adding to the momentum, and we can thus increase the extent and violence of the vibrations to any degree. We can stop the pendulum again by strokes applied when it is moving in the opposite direction, and the effects being added together will soon bring it to rest. Now if we alter the intervals of the strokes so that each two successive strokes act in opposite manners they will neutralise each other, and the energy expended will be turned into heat or sound at the point of percussion. Similar effects occur in all cases of rhythmical motion. If a musical note is sounded in a room containing a piano, the string corresponding to it will be thrown into vibration, because every successive stroke of the air-waves upon the string finds it in like position as regards the vibration, and thus adds to its energy of motion. But the other strings being incapable of vibrating with the same rapidity are struck at various points of their vibrations, and one stroke will soon be opposed by one contrary in effect. All phenomena of *resonance* arise from this coincidence in time of undulation. The air in a pipe closed at one end, and about 12 inches in length, is capable of vibrating 512 times in a second. If, then, the note C is sounded in front of the open end of the pipe, every successive vibration of the air is treasured up as it were in the motion of the air. In a pipe of different length the pulses of air would strike each other, and the mechanical energy being transmuted into heat would become no longer perceptible as sound.
Accumulated vibrations sometimes become so intense as to lead to unexpected results. A glass vessel if touched with a violin bow at a suitable point may be fractured with the violence of vibration. A suspension bridge may be broken down if a company of soldiers walk across it in steps the intervals of which agree with the vibrations of the bridge itself. But if they break the step or march in either quicker or slower pace, they may have no perceptible effect upon the bridge. In fact if the impulses communicated to any vibrating body are synchronous with its vibrations, the energy of those vibrations will be unlimited, and may fracture any body.
Let us now consider what will happen if the strokes be not exactly at the same intervals as the vibrations of the body, but, say, a little slower. Then a succession of strokes will meet the body in nearly but not quite the same position, and their efforts will be accumulated. Afterwards the strokes will begin to fall when the body is in the opposite phase. Imagine that one pendulum moving from one extreme point to another in a second, should be struck by another pendulum which makes 61 beats in a minute; then, if the pendulums commence together, they will at the end of 30-1/2 beats be moving in opposite directions. Hence whatever energy was communicated in the first half minute will be neutralised by the opposite effect of that given in the second half. The effect of the strokes of the second pendulum will therefore be alternately to increase and decrease the vibrations of the first, so that a new kind of vibration will be produced running through its phases in 61 seconds. An effect of this kind was actually observed by Ellicott, a member of the Royal Society, in the case of two clocks.[373] He found that through the wood-work by which the clocks were connected a slight impulse was transmitted, and each pendulum alternately lost and gained momentum. Each clock, in fact, tended to stop the other at regular intervals, and in the intermediate times to be stopped by the other.
[373] *Philosophical Transactions*, (1739), vol. xli. p. 126.Return to passage
Many disturbances in the planetary system depend upon the same principle; for if one planet happens always to pull another in the same direction in similar parts of their orbits, the effects, however slight, will be accumulated, and a disturbance of large ultimate amount and of long period will be produced. The long inequality in the motions of Jupiter and Saturn is thus due to the fact that five times the mean motion of Saturn is very nearly equal to twice the mean motion of Jupiter, causing a coincidence in their relative positions and disturbing powers. The rolling of ships depends mainly upon the question whether the period of vibration of the ship corresponds or not with the intervals at which the waves strike her. Much which seems at first sight unaccountable in the behaviour of vessels is thus explained, and the loss of the *Captain* is a sad case in point.
XXI. THEORY OF APPROXIMATION
In order that we may gain a true understanding of the kind, degree, and value of the knowledge which we acquire by experimental investigation, it is requisite that we should be fully conscious of its approximate character. We must learn to distinguish between what we can know and cannot know--between the questions which admit of solution, and those which only seem to be solved. Many persons may be misled by the expression *exact science*, and may think that the knowledge acquired by scientific methods admits of our reaching absolutely true laws, exact to the last degree. There is even a prevailing impression that when once mathematical formulæ have been successfully applied to a branch of science, this portion of knowledge assumes a new nature, and admits of reasoning of a higher character than those sciences which are still unmathematical.
The very satisfactory degree of accuracy attained in the science of astronomy gives a certain plausibility to erroneous notions of this kind. Some persons no doubt consider it to be *proved* that planets move in ellipses, in such a manner that all Kepler’s laws hold exactly true; but there is a double error in any such notions. In the first place, Kepler’s laws are *not proved*, if by proof we mean certain demonstration of their exact truth. In the next place, even assuming Kepler’s laws to be exactly true in a theoretical point of view, the planets never move according to those laws. Even if we could observe the motions of a planet, of a perfect globular form, free from all perturbing or retarding forces, we could never prove that it moved in a perfect ellipse. To prove the elliptical form we should have to measure infinitely small angles, and infinitely small fractions of a second; we should have to perform impossibilities. All we can do is to show that the motion of an unperturbed planet approaches *very nearly* to the form of an ellipse, and more nearly the more accurately our observations are made. But if we go on to assert that the path *is* an ellipse we pass beyond our data, and make an assumption which cannot be verified by observation.
But, secondly, as a matter of fact no planet does move in a perfect ellipse, or manifest the truth of Kepler’s laws exactly. The law of gravity prevents its own results from being clearly exhibited, because the mutual perturbations of the planets distort the elliptical paths. Those laws, again, hold exactly true only of infinitely small bodies, and when two great globes, like the sun and Jupiter, attract each other, the law must be modified. The periodic time is then shortened in the ratio of the square root of the number expressing the sun’s mass, to that of the sum of the numbers expressing the masses of the sun and planet, as was shown by Newton.[374] Even at the present day discrepancies exist between the observed dimensions of the planetary orbits and their theoretical magnitudes, after making allowance for all disturbing causes.[375] Nothing is more certain in scientific method than that approximate coincidence alone can be expected. In the measurement of continuous quantity perfect correspondence must be accidental, and should give rise to suspicion rather than to satisfaction.
[374] *Principia*, bk. iii. Prop. 15.
[375] Lockyer’s *Lessons in Elementary Astronomy*, p. 301.
One remarkable result of the approximate character of our observations is that we could never prove the existence of perfectly circular or parabolic movement, even if it existed. The circle is a singular case of the ellipse, for which the eccentricity is zero; it is infinitely improbable that any planet, even if undisturbed by other bodies, would have a circle for its orbit; but if the orbit were a circle we could never prove the entire absence of eccentricity. All that we could do would be to declare the divergence from the circular form to be inappreciable. Delambre was unable to detect the slightest ellipticity in the orbit of Jupiter’s first satellite, but he could only infer that the orbit was *nearly* circular. The parabola is the singular limit between the ellipse and the hyperbola. As there are elliptic and hyperbolic comets, so we might conceive the existence of a parabolic comet. Indeed if an undisturbed comet fell towards the sun from an infinite distance it would move in a parabola; but we could never prove that it so moved.
*Substitution of Simple Hypotheses.*
In truth men never can solve problems fulfilling the complex circumstances of nature. All laws and explanations are in a certain sense hypothetical, and apply exactly to nothing which we can know to exist. In place of the actual objects which we see and feel, the mathematician substitutes imaginary objects, only partially resembling those represented, but so devised that the discrepancies are not of an amount to alter seriously the character of the solution. When we probe the matter to the bottom physical astronomy is as hypothetical as Euclid’s elements. There may exist in nature perfect straight lines, triangles, circles, and other regular geometrical figures; to our science it is a matter of indifference whether they do or do not exist, because in any case they must be beyond our powers of perception. If we submitted a perfect circle to the most rigorous scrutiny, it is impossible that we should discover whether it were perfect or not. Nevertheless in geometry we argue concerning perfect curves, and rectilinear figures, and the conclusions apply to existing objects so far as we can assure ourselves that they agree with the hypothetical conditions of our reasoning. This is in reality all that we can do in the most perfect of the sciences.
Doubtless in astronomy we meet with the nearest approximation to actual conditions. The law of gravity is not a complex one in itself, and we believe it with much probability to be exactly true; but we cannot calculate out in any real case its accurate results. The law asserts that every particle of matter in the universe attracts every other particle, with a force depending on the masses of the particles and their distances. We cannot know the force acting on any particle unless we know the masses and distances and positions of all other particles in the universe. The physical astronomer has made a sweeping assumption, namely, that all the millions of existing systems exert no perturbing effects on our planetary system, that is to say, no effects in the least appreciable. The problem at once becomes hypothetical, because there is little doubt that gravitation between our sun and planets and other systems does exist. Even when they consider the relations of our planetary bodies *inter se*, all their processes are only approximate. In the first place they assume that each of the planets is a perfect ellipsoid, with a smooth surface and a homogeneous interior. That this assumption is untrue every mountain and valley, every sea, every mine affords conclusive evidence. If astronomers are to make their calculations perfect, they must not only take account of the Himalayas and the Andes, but must calculate separately the attraction of every hill, nay, of every ant-hill. So far are they from having considered any local inequality of the surface, that they have not yet decided upon the general form of the earth; it is still a matter of speculation whether or not the earth is an ellipsoid with three unequal axes. If, as is probable, the globe is irregularly compressed in some directions, the calculations of astronomers will have to be repeated and refined, in order that they may approximate to the attractive power of such a body. If we cannot accurately learn the form of our own earth, how can we expect to ascertain that of the moon, the sun, and other planets, in some of which probably are irregularities of greater proportional amount?
In a further way the science of physical astronomy is merely approximate and hypothetical. Given homogeneous ellipsoids acting upon each other according to the law of gravity, the best mathematicians have never and perhaps never will determine exactly the resulting movements. Even when three bodies simultaneously attract each other the complication of effects is so great that only approximate calculations can be made. Astronomers have not even attempted the general problem of the simultaneous attractions of four, five, six, or more bodies; they resolve the general problem into so many different problems of three bodies. The principle upon which the calculations of physical astronomy proceed, is to neglect every quantity which does not seem likely to lead to an effect appreciable in observation, and the quantities rejected are far more numerous and complex than the few larger terms which are retained. All then is merely approximate.
Concerning other branches of physical science the same statements are even more evidently true. We speak and calculate about inflexible bars, inextensible lines, heavy points, homogeneous substances, uniform spheres, perfect fluids and gases, and we deduce a great number of beautiful theorems; but all is hypothetical. There is no such thing as an inflexible bar, an inextensible line, nor any one of the other perfect objects of mechanical science; they are to be classed with those mythical existences, the straight line, triangle, circle, &c., about which Euclid so freely reasoned. Take the simplest operation considered in statics--the use of a crowbar in raising a heavy stone, and we shall find, as Thomson and Tait have pointed out, that we neglect far more than we observe.[376] If we suppose the bar to be quite rigid, the fulcrum and stone perfectly hard, and the points of contact real points, we may give the true relation of the forces. But in reality the bar must bend, and the extension and compression of different parts involve us in difficulties. Even if the bar be homogeneous in all its parts, there is no mathematical theory capable of determining with accuracy all that goes on; if, as is infinitely more probable, the bar is not homogeneous, the complete solution will be immensely more complicated, but hardly more hopeless. No sooner had we determined the change of form according to simple mechanical principles, than we should discover the interference of thermodynamic principles. Compression produces heat and extension cold, and thus the conditions of the problem are modified throughout. In attempting a fourth approximation we should have to allow for the conduction of heat from one part of the bar to another. All these effects are utterly inappreciable in a practical point of view, if the bar be a good stout one; but in a theoretical point of view they entirely prevent our saying that we have solved a natural problem. The faculties of the human mind, even when aided by the wonderful powers of abbreviation conferred by analytical methods, are utterly unable to cope with the complications of any real problem. And had we exhausted all the known phenomena of a mechanical problem, how can we tell that hidden phenomena, as yet undetected, do not intervene in the commonest actions? It is plain that no phenomenon comes within the sphere of our senses unless it possesses a momentum capable of irritating the appropriate nerves. There may then be worlds of phenomena too slight to rise within the scope of our consciousness.
[376] *Treatise on Natural Philosophy*, vol. i. pp. 337, &c.
All the instruments with which we perform our measurements are faulty. We assume that a plumb-line gives a vertical line; but this is never true in an absolute sense, owing to the attraction of mountains and other inequalities in the surface of the earth. In an accurate trigonometrical survey, the divergencies of the plumb-line must be approximately determined and allowed for. We assume a surface of mercury to be a perfect plane, but even in the breadth of 5 inches there is a calculable divergence from a true plane of about one ten-millionth part of an inch; and this surface further diverges from true horizontality as the plumb-line does from true verticality. That most perfect instrument, the pendulum, is not theoretically perfect, except for infinitely small arcs of vibration, and the delicate experiments performed with the torsion balance proceed on the assumption that the force of torsion of a wire is proportional to the angle of torsion, which again is only true for infinitely small angles.
Such is the purely approximate character of all our operations that it is not uncommon to find the theoretically worse method giving truer results than the theoretically perfect method. The common pendulum which is not isochronous is better for practical purposes than the cycloidal pendulum, which is isochronous in theory but subject to mechanical difficulties. The spherical form is not the correct form for a speculum or lense, but it differs so slightly from the true form, and is so much more easily produced mechanically, that it is generally best to rest content with the spherical surface. Even in a six-feet mirror the difference between the parabola and the sphere is only about one ten-thousandth part of an inch, a thickness which would be taken off in a few rubs of the polisher. Watts’ ingenious parallel motion was intended to produce rectilinear movement of the piston-rod. In reality the motion was always curvilinear, but for his purposes a certain part of the curve approximated sufficiently to a straight line.
*Approximation to Exact Laws.*
Though we can not prove numerical laws with perfect accuracy, it would be a great mistake to suppose that there is any inexactness in the laws of nature. We may even discover a law which we believe to represent the action of forces with perfect exactness. The mind may seem to pass in advance of its data, and choose out certain numerical results as absolutely true. We can never really pass beyond our data, and so far as assumption enters in, so far want of certainty will attach to our conclusions; nevertheless we may sometimes rightly prefer a probable assumption of a precise law to numerical results, which are at the best only approximate. We must accordingly draw a strong distinction between the laws of nature which we believe to be accurately stated in our formulas, and those to which our statements only make an approximation, so that at a future time the law will be differently stated.
The law of gravitation is expressed in the form F = Mm/D^{2}, meaning that gravity is proportional directly to the product of the gravitating masses, and indirectly to the square of their distance. The latent heat of steam is expressed by the equation log F = *a* + *b*α^{t} + *c*β^{t}, in which are five quantities *a*, *b*, *c*, α, β, to be determined by experiment. There is every reason to believe that in the progress of science the law of gravity will remain entirely unaltered, and the only effect of further inquiry will be to render it a more and more probable expression of the absolute truth. The law of the latent heat of steam on the other hand, will be modified by every new series of experiments, and it may not improbably be shown that the assumed law can never be made to agree exactly with the results of experiment.
Philosophers have not always supposed that the law of gravity was exactly true. Newton, though he had the highest confidence in its truth, admitted that there were motions in the planetary system which he could not reconcile with the law. Euler and Clairaut who were, with D’Alembert, the first to apply the full powers of mathematical analysis to the theory of gravitation as explaining the perturbations of the planets, did not think the law sufficiently established to attribute all discrepancies to the errors of calculation and observation. They did not feel certain that the force of gravity exactly obeyed the well-known rule. The law might involve other powers of the distance. It might be expressed in the form
F = ... + *a*/D + *b*/D^{2} + *c*/D^{3} + ...
and the coefficients *a* and *c* might be so small that those terms would become apparent only in very accurate comparisons with fact. Attempts have been made to account for difficulties, by attributing value to such neglected terms. Gauss at one time thought the even more fundamental principle of gravity, that the force is dependent only on mass and distance, might not be exactly true, and he undertook accurate pendulum experiments to test this opinion. Only as repeated doubts have time after time been resolved in favour of the law of Newton, has it been assumed as precisely correct. But this belief does not rest on experiment or observation only. The calculations of physical astronomy, however accurate, could never show that the other terms of the above expression were absolutely devoid of value. It could only be shown that they had such slight value as never to become apparent.
There are, however, other reasons why the law is probably complete and true as commonly stated. Whatever influence spreads from a point, and expands uniformly through space, will doubtless vary inversely in intensity as the square of the distance, because the area over which it is spread increases as the square of the radius. This part of the law of gravity may be considered as due to the properties of space, and there is a perfect analogy in this respect between gravity and all other *emanating* forces, as was pointed out by Keill.[377] Thus the undulations of light, heat, and sound, and the attractions of electricity and magnetism obey the very same law so far as we can ascertain. If the molecules of a gas or the particles of matter constituting odour were to start from a point and spread uniformly, their distances would increase and their density decrease according to the same principle.
[377] *An Introduction to Natural Philosophy*, 3rd edit. 1733, p. 5.
Other laws of nature stand in a similar position. Dalton’s laws of definite combining proportions never have been, and never can be, exactly proved; but chemists having shown, to a considerable degree of approximation, that the elements combine together as if each element had atoms of an invariable mass, assume that this is exactly true. They go even further. Prout pointed out in 1815 that the equivalent weights of the elements appeared to be simple numbers; and the researches of Dumas, Pelouze, Marignac, Erdmann, Stas, and others have gradually rendered it likely that the atomic weights of hydrogen, carbon, oxygen, nitrogen, chlorine, and silver, are in the ratios of the numbers 1, 12, 16, 14, 35·5, and 108. Chemists then step beyond their data; they throw aside their actual experimental numbers, and assume that the true ratios are not those exactly indicated by any weighings, but the simple ratios of these numbers. They boldly assume that the discrepancies are due to experimental errors, and they are justified by the fact that the more elaborate and skilful the researches on the subject, the more nearly their assumption is verified. Potassium is the only element whose atomic weight has been determined with great care, but which has not shown an approach to a simple ratio with the other elements. This exception may be due to some unsuspected cause of error.[378] A similar assumption is made in the law of definite combining volumes of gases, and Brodie has clearly pointed out the line of argument by which the chemist, observing that the discrepancies between the law and fact are within the limits of experimental error, assumes that they are due to error.[379]
[378] Watts, *Dictionary of Chemistry*, vol. i. p. 455.
[379] *Philosophical Transactions*, (1866) vol. clvi. p. 809.
Faraday, in one of his researches, expressly makes an assumption of the same kind. Having shown, with some degree of experimental precision, that there exists a simple proportion between quantities of electrical energy and the quantities of chemical substances which it can decompose, so that for every atom dissolved in the battery cell an atom ought theoretically, that is without regard to dissipation of some of the energy, to be decomposed in the electrolytic cell, he does not stop at his numerical results. “I have not hesitated,” he says,[380] “to apply the more strict results of chemical analysis to correct the numbers obtained as electrolytic results. This, it is evident, may be done in a great number of cases, without using too much liberty towards the due severity of scientific research.”
[380] *Experimental Researches in Electricity*, vol. i. p. 246.
The law of the conservation of energy, one of the widest of all physical generalisations, rests upon the same footing. The most that we can do by experiment is to show that the energy entering into any experimental combination is almost equal to what comes out of it, and more nearly so the more accurately we perform the measurements. Absolute equality is always a matter of assumption. We cannot even prove the indestructibility of matter; for were an exceedingly minute fraction of existing matter to vanish in any experiment, say one part in ten millions, we could never detect the loss.
*Successive Approximations to Natural Conditions.*
When we examine the history of scientific problems, we find that one man or one generation is usually able to make but a single step at a time. A problem is solved for the first time by making some bold hypothetical simplification, upon which the next investigator makes hypothetical modifications approaching more nearly to the truth. Errors are successively pointed out in previous solutions, until at last there might seem little more to be desired. Careful examination, however, will show that a series of minor inaccuracies remain to be corrected and explained, were our powers of reasoning sufficiently great, and the purpose adequate in importance.
Newton’s successful solution of the problem of the planetary movements entirely depended at first upon a great simplification. The law of gravity only applies directly to two infinitely small particles, so that when we deal with vast globes like the earth, Jupiter, and the sun, we have an immense aggregate of separate attractions to deal with, and the law of the aggregate need not coincide with the law of the elementary particles. But Newton, by a great effort of mathematical reasoning, was able to show that two homogeneous spheres of matter act as if the whole of their masses were concentrated at the centres; in short, that such spheres are centrobaric bodies (p. 364). He was then able with comparative ease to calculate the motions of the planets on the hypothesis of their being spheres, and to show that the results roughly agreed with observation. Newton, indeed, was one of the few men who could make two great steps at once. He did not rest contented with the spherical hypothesis; having reason to believe that the earth was really a spheroid with a protuberance around the equator, he proceeded to a second approximation, and proved that the attraction of the protuberant matter upon the moon accounted for the precession of the equinoxes, and led to various complicated effects. But, (p. 459), even the spheroidal hypothesis is far from the truth. It takes no account of the irregularities of surface, the great protuberance of land in Central Asia and South America, and the deficiency in the bed of the Atlantic.
To determine the law according to which a projectile, such as a cannon ball, moves through the atmosphere is a problem very imperfectly solved at the present day, but in which many successive advances have been made. So little was known concerning the subject three or four centuries ago that a cannon ball was supposed to move at first in a straight line, and after a time to be deflected into a curve. Tartaglia ventured to maintain that the path was curved throughout, as by the principle of continuity it should be; but the ingenuity of Galileo was required to prove this opinion, and to show that the curve was approximately a parabola. It is only, however, under forced hypotheses that we can assert the path of a projectile to be truly a parabola: the path must be through a perfect vacuum, where there is no resisting medium of any kind; the force of gravity must be uniform and act in parallel lines; or else the moving body must be either a mere point, or a perfect centrobaric body, that is a body possessing a definite centre of gravity. These conditions cannot be really fulfilled in practice. The next great step in the problem was made by Newton and Huyghens, the latter of whom asserted that the atmosphere would offer a resistance proportional to the velocity of the moving body, and concluded that the path would have in consequence a logarithmic character. Newton investigated in a general manner the subject of resisting media, and came to the conclusion that the resistance is more nearly proportional to the square of the velocity. The subject then fell into the hands of Daniel Bernoulli, who pointed out the enormous resistance of the air in cases of rapid movement, and calculated that a cannon ball, if fired vertically in a vacuum, would rise eight times as high as in the atmosphere. In recent times an immense amount both of theoretical and experimental investigation has been spent upon the subject, since it is one of importance in the art of war. Successive approximations to the true law have been made, but nothing like a complete and final solution has been achieved or even hoped for.[381]
[381] Hutton’s *Mathematical Dictionary*, vol. ii. pp. 287–292.
It is quite to be expected that the earliest experimenters in any branch of science will overlook errors which afterwards become most apparent. The Arabian astronomers determined the meridian by taking the middle point between the places of the sun when at equal altitudes on the same day. They overlooked the fact that the sun has its own motion in the time between the observations. Newton thought that the mutual disturbances of the planets might be disregarded, excepting perhaps the effect of the mutual attraction of the greater planets, Jupiter and Saturn, near their conjunction.[382] The expansion of quicksilver was long used as the measure of temperature, no clear idea being possessed of temperature apart from some of its more obvious effects. Rumford, in the first experiment leading to a determination of the mechanical equivalent of heat, disregarded the heat absorbed by the apparatus, otherwise he would, in Dr. Joule’s opinion, have come nearly to the correct result.
[382] *Principia*, bk. iii. Prop. 13.
It is surprising to learn the number of causes of error which enter into the simplest experiment, when we strive to attain rigid accuracy. We cannot accurately perform the simple experiment of compressing gas in a bent tube by a column of mercury, in order to test the truth of Boyle’s Law, without paying regard to--(1) the variations of atmospheric pressure, which are communicated to the gas through the mercury; (2) the compressibility of mercury, which causes the column of mercury to vary in density; (3) the temperature of the mercury throughout the column; (4) the temperature of the gas, which is with difficulty maintained invariable; (5) the expansion of the glass tube containing the gas. Although Regnault took all these circumstances into account in his examination of the law,[383] there is no reason to suppose that he exhausted the sources of inaccuracy.
[383] Jamin, *Cours de Physique*, vol. i. pp. 282, 283.
The early investigations concerning the nature of waves in elastic media proceeded upon the assumption that waves of different lengths would travel with equal speed. Newton’s theory of sound led him to this conclusion, and observation (p. 295) had verified the inference. When the undulatory theory came to be applied at the commencement of this century to explain the phenomena of light, a great difficulty was encountered. The angle at which a ray of light is refracted in entering a denser medium depends, according to that theory, on the velocity with which the wave travels, so that if all waves of light were to travel with equal velocity in the same medium, the dispersion of mixed light by the prism and the production of the spectrum could not take place. Some most striking phenomena were thus in direct conflict with the theory. Cauchy first pointed out the explanation, namely, that all previous investigators had made an arbitrary assumption for the sake of simplifying the calculations. They had assumed that the particles of the vibrating medium are so close together that the intervals are inconsiderable compared with the length of the wave. This hypothesis happened to be approximately true in the case of air, so that no error was discovered in experiments on sound. Had it not been so, the earlier analysts would probably have failed to give any solution, and the progress of the subject might have been retarded. Cauchy was able to make a new approximation under the more difficult supposition, that the particles of the vibrating medium are situated at considerable distances, and act and react upon the neighbouring particles by attractive and repulsive forces. To calculate the rate of propagation of disturbance in such a medium is a work of excessive difficulty. The complete solution of the problem appears indeed to be beyond human power, so that we must be content, as in the case of the planetary motions, to look forward to successive approximations. All that Cauchy could do was to show that certain quantities, neglected in previous theories, became of considerable amount under the new conditions of the problem, so that there will exist a relation between the length of the wave, and the velocity at which it travels. To remove, then, the difficulties in the way of the undulatory theory of light, a new approach to probable conditions was needed.[384]
[384] Lloyd’s *Lectures on the Wave Theory*, pp. 22, 23.
In a similar manner Fourier’s theory of the conduction and radiation of heat was based upon the hypothesis that the quantity of heat passing along any line is simply proportional to the rate of change of temperature. But it has since been shown by Forbes that the conductivity of a body diminishes as its temperature increases. All the details of Fourier’s solution therefore require modification, and the results are in the meantime to be regarded as only approximately true.[385]
[385] Tait’s *Thermodynamics*, p. 10.
We ought to distinguish between those problems which are physically and those which are merely mathematically incomplete. In the latter case the physical law is correctly seized, but the mathematician neglects, or is more often unable to follow out the law in all its results. The law of gravitation and the principles of harmonic or undulatory movement, even supposing the data to be correct, can never be followed into all their ultimate results. Young explained the production of Newton’s rings by supposing that the rays reflected from the upper and lower surfaces of a thin film of a certain thickness were in opposite phases, and thus neutralised each other. It was pointed out, however, that as the light reflected from the nearer surface must be undoubtedly a little brighter than that from the further surface, the two rays ought not to neutralise each other so completely as they are observed to do. It was finally shown by Poisson that the discrepancy arose only from incomplete solution of the problem; for the light which has once got into the film must be to a certain extent reflected backwards and forwards *ad infinitum*; and if we follow out this course of the light by perfect mathematical analysis, absolute darkness may be shown to result from the interference of the rays.[386] In this case the natural laws concerned, those of reflection and refraction, are accurately known, and the only difficulty consists in developing their full consequences.
[386] Lloyd’s *Lectures on the Wave Theory*, pp. 82, 83.
*Discovery of Hypothetically Simple Laws.*
In some branches of science we meet with natural laws of a simple character which are in a certain point of view exactly true and yet can never be manifested as exactly true in natural phenomena. Such, for instance, are the laws concerning what is called a *perfect gas*. The gaseous state of matter is that in which the properties of matter are exhibited in the simplest manner. There is much advantage accordingly in approaching the question of molecular mechanics from this side. But when we ask the question--What is a gas? the answer must be a hypothetical one. Finding that gases *nearly* obey the law of Boyle and Mariotte; that they *nearly* expand by heat at the uniform rate of one part in 272·9 of their volume at 0° for each degree centigrade; and that they *more nearly* fulfil these conditions the more distant the point of temperature at which we examine them from the liquefying point, we pass by the principle of continuity to the conception of a perfect gas. Such a gas would probably consist of atoms of matter at so great a distance from each other as to exert no attractive forces upon each other; but for this condition to be fulfilled the distances must be infinite, so that an absolutely perfect gas cannot exist. But the perfect gas is not merely a limit to which we may approach, it is a limit passed by at least one real gas. It has been shown by Despretz, Pouillet, Dulong, Arago, and finally Regnault, that all gases diverge from the Boylean law, and in nearly all cases the density of the gas increases in a somewhat greater ratio than the pressure, indicating a tendency on the part of the molecules to approximate of their own accord. In the more condensable gases such as sulphurous acid, ammonia, and cyanogen, this tendency is strongly apparent near the liquefying point. Hydrogen, on the contrary, diverges from the law of a perfect gas in the opposite direction, that is, the density increases less than in the ratio of the pressure.[387] This is a singular exception, the bearing of which I am unable to comprehend.
[387] Jamin, *Cours de Physique*, vol. i. pp. 283–288.
All gases diverge again from the law of uniform expansion by heat, but the divergence is less as the gas in question is less condensable, or examined at a temperature more removed from its liquefying point. Thus the perfect gas must have an infinitely high temperature. According to Dalton’s law each gas in a mixture retains its own properties unaffected by the presence of any other gas.[388] This law is probably true only by approximation, but it is obvious that it would be true of the perfect gas with infinitely distant particles.[389]
[388] Joule and Thomson, *Philosophical Transactions*, 1854, vol. cxliv. p. 337.
[389] The properties of a perfect gas have been described by Rankine, *Transactions of the Royal Society of Edinburgh*, vol. xxv. p. 561.
*Mathematical Principles of Approximation.*
The approximate character of physical science will be rendered more plain if we consider it from a mathematical point of view. Throughout quantitative investigations we deal with the relation of one quantity to other quantities, of which it is a function; but the subject is sufficiently complicated if we view one quantity as a function of one other. Now, as a general rule, a function can be developed or expressed as the sum of quantities, the values of which depend upon the successive powers of the variable quantity. If *y* be a function of *x* then we may say that
*y* = A + B*x* + C*x*^{2} + D*x*^{3} + E*x*^{4} ....
In this equation, A, B, C, D, &c., are fixed quantities, of different values in different cases. The terms may be infinite in number or after a time may cease to have any value. Any of the coefficients A, B, C, &c., may be zero or negative; but whatever they be they are fixed. The quantity *x* on the other hand may be made what we like, being variable. Suppose, in the first place, that *x* and *y* are both lengths. Let us assume that 1/10,000 part of an inch is the least that we can take note of. Then when *x* is one hundredth of an inch, we have *x*^{2} = 1/10,000, and if C be less than unity, the term C*x*^{2} will be inappreciable, being less than we can measure. Unless any of the quantities D, E, &c., should happen to be very great, it is evident that all the succeeding terms will also be inappreciable, because the powers of *x* become rapidly smaller in geometrical ratio. Thus when *x* is made small enough the quantity *y* seems to obey the equation
*y* = A + B*x*.
If *x* should be still less, if it should become as small, for instance, as 1/1,000,000 of an inch, and B should not be very great, then *y* would appear to be the fixed quantity A, and would not seem to vary with *x* at all. On the other hand, were x to grow greater, say equal to 1/10 inch, and C not be very small, the term C*x*^{2} would become appreciable, and the law would now be more complicated.
We can invert the mode of viewing this question, and suppose that while the quantity *y* undergoes variations depending on many powers of *x*, our power of detecting the changes of value is more or less acute. While our powers of observation remain very rude we may be unable to detect any change in the quantity at all, that is to say, B*x* may always be too small to come within our notice, just as in former days the fixed stars were so called because they remained at apparently fixed distances from each other. With the use of telescopes and micrometers we become able to detect the existence of some motion, so that the distance of one star from another may be expressed by A + B*x*, the term including *x*^{2} being still inappreciable. Under these circumstances the star will seem to move uniformly, or in simple proportion to the time *x*. With much improved means of measurement it will probably be found that this uniformity of motion is only apparent, and that there exists some acceleration or retardation. More careful investigation will show the law to be more and more complicated than was previously supposed.
There is yet another way of explaining the apparent results of a complicated law. If we take any curve and regard a portion of it free from any kind of discontinuity, we may represent the character of such portion by an equation of the form
*y* = A + B*x* + C*x*^{2} + D*x*^{3} + ....
Restrict the attention to a very small portion of the curve, and the eye will be unable to distinguish its difference from a straight line, which amounts to saying that in the portion examined the term C*x*^{2} has no value appreciable by the eye. Take a larger portion of the curve and it will be apparent that it possesses curvature, but it will be possible to draw a parabola or ellipse so that the curve shall apparently coincide with a portion of that parabola or ellipse. In the same way if we take larger and larger arcs of the curve it will assume the character successively of a curve of the third, fourth, and perhaps higher degrees; that is to say, it corresponds to equations involving the third, fourth, and higher powers of the variable quantity.
We have arrived then at the conclusion that every phenomenon, when its amount can only be rudely measured, will either be of fixed amount, or will seem to vary uniformly like the distance between two inclined straight lines. More exact measurement may show the error of this first assumption, and the variation will then appear to be like that of the distance between a straight line and a parabola or ellipse. We may afterwards find that a curve of the third or higher degrees is really required to represent the variation. I propose to call the variation of a quantity *linear*, *elliptic*, *cubic*, *quartic*, *quintic*, &c., according as it is discovered to involve the first, second, third, fourth, fifth, or higher powers of the variable. It is a general rule in quantitative investigation that we commence by discovering linear, and afterwards proceed to elliptic or more complicated laws of variation. The approximate curves which we employ are all, according to De Morgan’s use of the name, parabolas of some order or other; and since the common parabola of the second order is approximately the same as a very elongated ellipse, and is in fact an infinitely elongated ellipse, it is convenient and proper to call variation of the second order *elliptic*. It might also be called *quadric* variation.
As regards many important phenomena we are yet only in the first stage of approximation. We know that the sun and many so-called fixed stars, especially 61 Cygni, have a proper motion through space, and the direction of this motion at the present time is known with some degree of accuracy. But it is hardly consistent with the theory of gravity that the path of any body should really be a straight line. Hence, we must regard a rectilinear path as only a provisional description of the motion, and look forward to the time when its curvature will be detected, though centuries perhaps must first elapse.
We are accustomed to assume that on the surface of the earth the force of gravity is uniform, because the variation is of so slight an amount that we are scarcely able to detect it. But supposing we could measure the variation, we should find it simply proportional to the height. Taking the earth’s radius to be unity, let *h* be the height at which we measure the force of gravity. Then by the well-known law of the inverse square, that force will be proportional to
*g*/(1 + *h*)^{2}, or to *g*(1 - 2*h* + 3*h*^{2} - 4*h*^{3} + ...).
But at all heights to which we can attain *h* will be so small a fraction of the earth’s radius that 3*h*^{2} will be inappreciable, and the force of gravity will seem to follow the law of linear variation, being proportional to 1 - 2*h*.
When the circumstances of an experiment are much altered, different powers of the variable may become prominent. The resistance of a liquid to a body moving through it may be approximately expressed as the sum of two terms respectively involving the first and second powers of the velocity. At very low velocities the first power is of most importance, and the resistance, as Professor Stokes has shown, is nearly in simple proportion to the velocity. When the motion is rapid the resistance increases in a still greater degree, and is more nearly proportional to the square of the velocity.
*Approximate Independence of Small Effects.*
One result of the theory of approximation possesses such importance in physical science, and is so often applied, that we may consider it separately. The investigation of causes and effects is immensely simplified when we may consider each cause as producing its own effect invariably, whether other causes are acting or not. Thus, if the body P produces *x*, and Q produces *y*, the question is whether P and Q acting together will produce the sum of the separate effects, *x* + *y*. It is under this supposition that we treated the methods of eliminating error (Chap. XV.), and errors of a less amount would still remain if the supposition was a forced one. There are probably some parts of science in which the supposition of independence of effects holds rigidly true. The mutual gravity of two bodies is entirely unaffected by the presence of other gravitating bodies. People do not usually consider that this important principle is involved in such a simple thing as putting two pound weights in the scale of a balance. How do we know that two pounds together will weigh twice as much as one? Do we know it to be exactly so? Like other results founded on induction we cannot prove it absolutely, but all the calculations of physical astronomy proceed upon the assumption, so that we may consider it proved to a very high degree of approximation. Had not this been true, the calculations of physical astronomy would have been infinitely more complex than they actually are, and the progress of knowledge would have been much slower.
It is a general principle of scientific method that if effects be of small amount, comparatively to our means of observation, all joint effects will be of a higher order of smallness, and may therefore be rejected in a first approximation. This principle was employed by Daniel Bernoulli in the theory of sound, under the title of *The Principle of the Coexistence of Small Vibrations*. He showed that if a string is affected by two kinds of vibrations, we may consider each to be going on as if the other did not exist. We cannot perceive that the sounding of one musical instrument prevents or even modifies the sound of another, so that all sounds would seem to travel through the air, and act upon the ear in independence of each other. A similar assumption is made in the theory of tides, which are great waves. One wave is produced by the attraction of the moon, and another by the attraction of the sun, and the question arises, whether when these waves coincide, as at the time of spring tides, the joint wave will be simply the sum of the separate waves. On the principle of Bernoulli this will be so, because the tides on the ocean are very small compared with the depth of the ocean.
The principle of Bernoulli, however, is only approximately true. A wave never is exactly the same when another wave is interfering with it, but the less the displacement of particles due to each wave, the less in a still higher degree is the effect of one wave upon the other. In recent years Helmholtz was led to suspect that some of the phenomena of sound might after all be due to resultant effects overlooked by the assumption of previous physicists. He investigated the secondary waves which would arise from the interference of considerable disturbances, and was able to show that certain summation of resultant tones ought to be heard, and experiments subsequently devised for the purpose showed that they might be heard.
Throughout the mechanical sciences the *Principle of the Superposition of Small Motions* is of fundamental importance,[390] and it may be thus explained. Suppose that two forces, acting from the points B and C, are simultaneously moving a body A. Let the force acting from B be such that in one second it would move A to *p*, and similarly let the second force, acting alone, move A to *r*. The question arises, then, whether their joint action will urge A to *q* along the diagonal of the parallelogram. May we say that A will move the distance A*p* in the direction AB, and A*r* in the direction AC, or, what is the same thing, along the parallel line *pq*? In strictness we cannot say so; for when A has moved towards *p*, the force from C will no longer act along the line AC, and similarly the motion of A towards *r* will modify the action of the force from B. This interference of one force with the line of action of the other will evidently be greater the larger is the extent of motion considered; on the other hand, as we reduce the parallelogram A*pqr*, compared with the distances AB and AC, the less will be the interference of the forces. Accordingly mathematicians avoid all error by considering the motions as infinitely small, so that the interference becomes of a still higher order of infinite smallness, and may be entirely neglected. By the resources of the differential calculus it is possible to calculate the motion of the particle A, as if it went through an infinite number of infinitely small diagonals of parallelograms. The great discoveries of Newton really arose from applying this method of calculation to the movements of the moon round the earth, which, while constantly tending to move onward in a straight line, is also deflected towards the earth by gravity, and moves through an elliptic curve, composed as it were of the infinitely small diagonals of infinitely numerous parallelograms. The mathematician, in his investigation of a curve, always treats it as made up of a great number of straight lines, and it may be doubted whether he could treat it in any other manner. There is no error in the final results, because having obtained the formulæ flowing from this supposition, each straight line is then regarded as becoming infinitely small, and the polygonal line becomes undistinguishable from a perfect curve.[391]
[390] Thomson and Tait’s *Natural Philosophy*, vol. i. p. 60.
[391] Challis, *Notes on the Principles of Pure and Applied Calculation*, 1869, p. 83.
In abstract mathematical theorems the approximation to absolute truth is perfect, because we can treat of infinitesimals. In physical science, on the contrary, we treat of the least quantities which are perceptible. Nevertheless, while carefully distinguishing between these two different cases, we may fearlessly apply to both the principle of the superposition of small effects. In physical science we have only to take care that the effects really are so small that any joint effect will be unquestionably imperceptible. Suppose, for instance, that there is some cause which alters the dimensions of a body in the ratio of 1 to 1 + α, and another cause which produces an alteration in the ratio of 1 to 1 + β. If they both act at once the change will be in the ratio of 1 to (1 + α)(1 + β), or as 1 to 1 + α + β + αβ. But if α and β be both very small fractions of the total dimensions, αβ will be yet far smaller and may be disregarded; the ratio of change is then approximately that of 1 to 1 + α + β, or the joint effect is the sum of the separate effects. Thus if a body were subjected to three strains, at right angles to each other, the total change in the volume of the body would be approximately equal to the sum of the changes produced by the separate strains, provided that these are very small. In like manner not only is the expansion of every solid and liquid substance by heat approximately proportional to the change of temperature, when this change is very small in amount, but the cubic expansion may also be considered as being three times as great as the linear expansion. For if the increase of temperature expands a bar of metal in the ratio of 1 to 1 + α, and the expansion be equal in all directions, then a cube of the same metal would expand as 1 to (1 + α)^{3}, or as 1 to 1 + 3α + 3α^{2} + α^{3}. When α is a very small quantity the third term 3α^{2} will be imperceptible, and still more so the fourth term α^{3}. The coefficients of expansion of solids are in fact so small, and so imperfectly determined, that physicists seldom take into account their second and higher powers.
It is a result of these principles that all small errors may be assumed to vary in simple proportion to their causes--a new reason why, in eliminating errors, we should first of all make them as small as possible. Let us suppose that there is a right-angled triangle of which the two sides containing the right angle are really of the lengths 3 and 4, so that the hypothenuse is √(3^{2} + 4^{2}) or 5. Now, if in two measurements of the first side we commit slight errors, making it successively 4·001 and 4·002, then calculation will give the lengths of the hypothenuse as almost exactly 5·0008 and 5·0016, so that the error in the hypothenuse will seem to vary in simple proportion to that of the side, although it does not really do so with perfect exactness. The logarithm of a number does not vary in proportion to that number--nevertheless we find the difference between the logarithms of the numbers 100000 and 100001 to be almost exactly equal to that between the numbers 100001 and 100002. It is thus a general rule that very small differences between successive values of a function are approximately proportional to the small differences of the variable quantity.
On these principles it is easy to draw up a series of rules such as those given by Kohlrausch[392] for performing calculations in an abbreviated form when the variable quantity is very small compared with unity. Thus for 1 ÷ (1 + α) we may substitute 1 - α; for 1 ÷ (1 - α) we may put 1 + α; 1 ÷ √(1 + α) becomes 1 - (1/2)α, and so forth.
[392] *An Introduction to Physical Measurements*, translated by Waller and Procter, 1873, p. 10.
*Four Meanings of Equality.*
Although it might seem that there are few terms more free from ambiguity than the term *equal*, yet scientific men do employ it with at least four meanings, which it is desirable to distinguish. These meanings I may describe as
(1) Absolute Equality. (2) Sub-equality. (3) Apparent Equality. (4) Probable Equality.
By *absolute equality* we signify that which is complete and perfect to the last degree; but it is obvious that we can only know such equality in a theoretical or hypothetical manner. The areas of two triangles standing upon the same base and between the same parallels are absolutely equal. Hippocrates beautifully proved that the area of a lunula or figure contained between two segments of circles was absolutely equal to that of a certain right-angled triangle. As a general rule all geometrical and other elementary mathematical theorems involve absolute equality.
De Morgan proposed to describe as *sub-equal* those quantities which are equal within an infinitely small quantity, so that *x* is sub-equal to *x* + *dx*. The differential calculus may be said to arise out of the neglect of infinitely small quantities, and in mathematical science other subtle distinctions may have to be drawn between kinds of equality, as De Morgan has shown in a remarkable memoir “On Infinity; and on the sign of Equality.”[393]
[393] *Cambridge Philosophical Transactions* (1865), vol. xi. Part I.
*Apparent equality* is that with which physical science deals. Those magnitudes are apparently equal which differ only by an imperceptible quantity. To the carpenter anything less than the hundredth part of an inch is non-existent; there are few arts or artists to which the hundred-thousandth of an inch is of any account. Since all coincidence between physical magnitudes is judged by one or other sense, we must be restricted to a knowledge of apparent equality.
In reality even apparent equality is rarely to be expected. More commonly experiments will give only *probable equality*, that is results will come so near to each other that the difference may be ascribed to unimportant disturbing causes. Physicists often assume quantities to be equal provided that they fall within the limits of probable error of the processes employed. We cannot expect observations to agree with theory more closely than they agree with each other, as Newton remarked of his investigations concerning Halley’s Comet.
*Arithmetic of Approximate Quantities.*
Considering that almost all the quantities which we treat in physical and social science are approximate only, it seems desirable that attention should be paid in the teaching of arithmetic to the correct interpretation and treatment of approximate numerical statements. We seem to need notation for expressing the approximateness or exactness of decimal numbers. The fraction ·025 may mean either precisely one 40th part, or it may mean anything between ·0245 and ·0255. I propose that when a decimal fraction is completely and exactly given, a *small cipher* or circle should be added to indicate that there is nothing more to come, as in ·025◦. When the first figure of the decimals rejected is 5 or more, the first figure retained should be raised by a unit, according to a rule approved by De Morgan, and now generally recognised. To indicate that the fraction thus retained is more than the truth, a point has been placed over the last figure in some tables of logarithms; but a similar point is used to denote the period of a repeating decimal, and I should therefore propose to employ a colon *after* the figure; thus ·025: would mean that the true quantity lies between ·0245° and ·025° inclusive of the lower but not the higher limit. When the fraction is less than the truth, two dots might be placed horizontally as in 025.. which would mean anything between ·025° and ·0255° not inclusive.
When approximate numbers are added, subtracted, multiplied, or divided, it becomes a matter of some complexity to determine the degree of accuracy of the result. There are few persons who could assert off-hand that the sum of the approximate numbers 34·70, 52·693, 80·1, is 167·5 *within less than* ·07. Mr. Sandeman has traced out the rules of approximate arithmetic in a very thorough manner, and his directions are worthy of careful attention.[394] The third part of Sonnenschein and Nesbitt’s excellent book on arithmetic[395] describes fully all kinds of approximate calculations, and shows both how to avoid needless labour and how to take proper account of inaccuracy in operating with approximate decimal fractions. A simple investigation of the subject is to be found in Sonnet’s *Algèbre Elémentaire* (Paris, 1848) chap. xiv., “Des Approximations Absolues et Relatives.” There is also an American work on the subject.[396]
[394] Sandeman, *Pelicotetics*, p. 214.
[395] *The Science and Art of Arithmetic for the Use of Schools.* (Whitaker and Co.)
[396] *Principles of Approximate Calculations*, by J. J. Skinner, C.E. (New York, Henry Holt), 1876.Return to passage
Although the accuracy of measurement has so much advanced since the time of Leslie, it is not superfluous to repeat his protest against the unfairness of affecting by a display of decimal fractions a greater degree of accuracy than the nature of the case requires and admits.[397] I have known a scientific man to register the barometer to a second of time when the nearest quarter of an hour would have been amply sufficient. Chemists often publish results of analysis to the ten-thousandth or even the millionth part of the whole, when in all probability the processes employed cannot be depended on beyond the hundredth part. It is seldom desirable to give more than one place of figures of uncertain amount; but it must be allowed that a nice perception of the degree of accuracy possible and desirable is requisite to save misapprehension and needless computation on the one hand, and to secure all attainable exactness on the other hand.
[397] Leslie, *Inquiry into the Nature of Heat*, p. 505.Return to passage
XXII. QUANTITATIVE INDUCTION
We have not yet formally considered any processes of reasoning which have for their object to disclose laws of nature expressed in quantitative equations. We have been inquiring into the modes by which a phenomenon may be measured, and, if it be a composite phenomenon, may be resolved, by the aid of several measurements, into its component parts. We have also considered the precautions to be taken in the performance of observations and experiments in order that we may know what phenomena we really do measure, but we must remember that, no number of facts and observations can by themselves constitute science. Numerical facts, like other facts, are but the raw materials of knowledge, upon which our reasoning faculties must be exerted in order to draw forth the principles of nature. It is by an inverse process of reasoning that we can alone discover the mathematical laws to which varying quantities conform. By well-conducted experiments we gain a series of values of a variable, and a corresponding series of values of a variant, and we now want to know what mathematical function the variant is as regards the variable. In the usual progress of a science three questions will have to be answered as regards every important quantitative phenomenon:--
(1) Is there any constant relation between a variable and a variant?
(2) What is the empirical formula expressing this relation?
(3) What is the rational formula expressing the law of nature involved?
*Probable Connection of Varying Quantities.*
We find it stated by Mill,[398] that “Whatever phenomenon varies in any manner whenever another phenomenon varies in some particular manner, is either a cause or an effect of that phenomenon, or is connected with it through some fact of causation.” This assertion may be considered true when it is interpreted with sufficient caution; but it might otherwise lead us into error. There is nothing whatever in the nature of things to prevent the existence of two variations which should apparently follow the same law, and yet have no connection with each other. One binary star might be going through a revolution which, so far as we could tell, was of equal period with that of another binary star, and according to the above rule the motion of one would be the cause of the motion of the other, which would not be really the case. Two astronomical clocks might conceivably be made so nearly perfect that, for several years, no difference could be detected, and we might then infer that the motion of one clock was the cause or effect of the motion of the other. This matter requires careful discrimination. We must bear in mind that the continuous quantities of space, time, force, &c., which we measure, are made up of an infinite number of infinitely small units. We may then meet with two variable phenomena which follow laws so nearly the same, that in no part of the variations open to our observation can any discrepancy be discovered. I grant that if two clocks could be shown to have kept *exactly* the same time during any finite interval, the probability would become infinitely high that there was a connection between their motions. But we can never absolutely prove such coincidences to exist. Allow that we may observe a difference of one-tenth of a second in their time, yet it is possible that they were independently regulated so as to go together within less than that quantity of time. In short, it would require either an infinitely long time of observation, or infinitely acute powers of measuring discrepancy, to decide positively whether two clocks were or were not in relation with each other.
[398] *System of Logic*, bk. iii. chap. viii § 6.
A similar question actually occurs in the case of the moon’s motion. We have no record that any other portion of the moon was ever visible to men than such as we now see. This fact sufficiently proves that within the historical period the rotation of the moon on its own axis has coincided with its revolutions round the earth. Does this coincidence prove a relation of cause and effect to exist? The answer must be in the negative, because there might have been so slight a discrepancy between the motions that there has not yet been time to produce any appreciable effect. There may nevertheless be a high probability of connection.
The whole question of the relation of quantities thus resolves itself into one of probability. When we can only rudely measure a quantitative result, we can assign but slight importance to any correspondence. Because the brightness of two stars seems to vary in the same manner, there is no considerable probability that they have any relation with each other. Could it be shown that their periods of variation were the same to infinitely small quantities it would be certain, that is infinitely probable, that they were connected, however unlikely this might be on other grounds. The general mode of estimating such probabilities is identical with that applied to other inductive problems. That any two periods of variation should by chance become *absolutely equal* is infinitely improbable; hence if, in the case of the moon or other moving bodies, we could prove absolute coincidence we should have certainty of connection.[399] With approximate measurements, which alone are within our power, we must hope for approximate certainty at the most.
[399] Laplace, *System of the World*, translated by Harte, vol. ii. p. 366.
The principles of inference and probability, according to which we treat causes and effects varying in amount, are exactly the same as those by which we treated simple experiments. Continuous quantity, however, affords us an infinitely more extensive sphere of observation, because every different amount of cause, however little different, ought to be followed by a different amount of effect. If we can measure temperature to the one-hundredth part of a degree centigrade, then between 0° and 100° we have 10,000 possible trials. If the precision of our measurements is increased, so that the one-thousandth part of a degree can be appreciated, our trials may be increased tenfold. The probability of connection will be proportional to the accuracy of our measurements.
When we can vary the quantity of a cause at will it is easy to discover whether a certain effect is due to that cause or not. We can then make as many irregular changes as we like, and it is quite incredible that the supposed effect should by chance go through exactly the corresponding series of changes except by dependence. If we have a bell ringing *in vacuo*, the sound increases as we let in the air, and it decreases again as we exhaust the air. Tyndall’s singing flames evidently obeyed the directions of his own voice; and Faraday when he discovered the relation of magnetism and light found that, by making or breaking or reversing the current of the electro-magnet, he had complete command over a ray of light, proving beyond all reasonable doubt the dependence of cause and effect. In such cases it is the perfect coincidence in time between the change in the effect and that in the cause which raises a high improbability of casual coincidence.
It is by a simple case of variation that we infer the existence of a material connection between two bodies moving with exactly equal velocity, such as the locomotive engine and the train which follows it. Elaborate observations were requisite before astronomers could all be convinced that the red hydrogen flames seen during solar eclipses belonged to the sun, and not to the moon’s atmosphere as Flamsteed assumed. As early as 1706, Stannyan noticed a blood-red streak in an eclipse which he witnessed at Berne, and he asserted that it belonged to the sun; but his opinion was not finally established until photographs of the eclipse in 1860, taken by Mr. De la Rue, showed that the moon’s dark body gradually covered the red prominences on one side, and uncovered those on the other; in short, that these prominences moved precisely as the sun moved, and not as the moon moved.
Even when we have no means of accurately measuring the variable quantities we may yet be convinced of their connection, if one always varies perceptibly at the same time as the other. Fatigue increases with exertion; hunger with abstinence from food; desire and degree of utility decrease with the quantity of commodity consumed. We know that the sun’s heating power depends upon his height of the sky; that the temperature of the air falls in ascending a mountain; that the earth’s crust is found to be perceptibly warmer as we sink mines into it; we infer the direction in which a sound comes from the change of loudness as we approach or recede. The facility with which we can time after time observe the increase or decrease of one quantity with another sufficiently shows the connection, although we may be unable to assign any precise law of relation. The probability in such cases depends upon frequent coincidence in time.
*Empirical Mathematical Laws.*
It is important to acquire a clear comprehension of the part which is played in scientific investigation by empirical formulæ and laws. If we have a table containing certain values of a variable and the corresponding values of the variant, there are mathematical processes by which we can infallibly discover a mathematical formula yielding numbers in more or less exact agreement with the table. We may generally assume that the quantities will approximately conform to a law of the form
*y* = A + B*x* + C*x*^{2},
in which *x* is the variable and *y* the variant. We can then select from the table three values of *y*, and the corresponding values of *x*; inserting them in the equation, we obtain three equations by the solution of which we gain the values of A, B, and C. It will be found as a general rule that the formula thus obtained yields the other numbers of the table to a considerable degree of approximation.
In many cases even the second power of the variable will be unnecessary; Regnault found that the results of his elaborate inquiry into the latent heat of steam at different pressures were represented with sufficient accuracy by the empirical formula
λ = 606·5 + 0·305 *t*,
in which λ is the total heat of the steam, and *t* the temperature.[400] In other cases it may be requisite to include the third power of the variable. Thus physicists assume the law of the dilatation of liquids to be of the form
δ_{t} = *at* + *bt*^{2} + *ct*^{3},
[400] *Chemical Reports and Memoirs*, Cavendish Society, p. 294.
and they calculate from results of observation the values of the three constants *a*, *b*, *c*, which are usually small quantities not exceeding one-hundredth part of a unit, but requiring to be determined with great accuracy.[401] Theoretically speaking, this process of empirical representation might be applied with any degree of accuracy; we might include still higher powers in the formula, and with sufficient labour obtain the values of the constants, by using an equal number of experimental results. The method of least squares may also be employed to obtain the most probable values of the constants.
[401] Jamin, *Cours de Physique*, vol. ii. p. 38.
In a similar manner all periodic variations may be represented with any required degree of accuracy by formulæ involving the sines and cosines of angles and their multiples. The form of any tidal or other wave may thus be expressed, as Sir G. B. Airy has explained.[402] Almost all the phenomena registered by meteorologists are periodic in character, and when freed from disturbing causes may be embodied in empirical formulæ. Bessel has given a rule by which from any regular series of observations we may, on the principle of the method of least squares, calculate out with a moderate amount of labour a formula expressing the variation of the quantity observed, in the most probable manner. In meteorology three or four terms are usually sufficient for representing any periodic phenomenon, but the calculation might be carried to any higher degree of accuracy. As the details of the process have been described by Herschel in his treatise on Meteorology,[403] I need not further enter into them.
[402] *On Tides and Waves*, Encyclopædia Metropolitana, p. 366*.
[403] *Encyclopædia Britannica*, art. *Meteorology*. Reprint, §§ 152–156.
The reader might be tempted to think that in these processes of calculation we have an infallible method of discovering inductive laws, and that my previous statements (Chap. VII.) as to the purely tentative and inverse character of the inductive process are negatived. Were there indeed any general method of inferring laws from facts it would overturn my statement, but it must be carefully observed that these empirical formulæ do not coincide with natural laws. They are only approximations to the results of natural laws founded upon the general principles of approximation. It has already been pointed out that however complicated be the nature of a curve, we may examine so small a portion of it, or we may examine it with such rude means of measurement, that its divergence from an elliptic curve will not be apparent. As a still ruder approximation a portion of a straight line will always serve our purpose; but if we need higher precision a curve of the third or fourth degree will almost certainly be sufficient. Now empirical formulæ really represent these approximate curves, but they give us no information as to the precise nature of the curve itself to which we are approximating. We do not learn what function the variant is of the variable, but we obtain another function which, within the bounds of observation, gives nearly the same values.
*Discovery of Rational Formulæ.*
Let us now proceed to consider the modes in which from numerical results we can establish the actual relation between the quantity of the cause and that of the effect. What we want is a *rational* formula or function, which will exhibit the *reason* or exact nature and origin of the law in question. There is no word more frequently used by mathematicians than the word *function*, and yet it is difficult to define its meaning with perfect accuracy. Originally it meant performance or execution, being equivalent to the Greek λειτουργία or τέλεσμα. Mathematicians at first used it to mean *any power of a quantity*, but afterwards generalised it so as to include “any quantity formed in any manner whatsoever from another quantity.”[404] Any quantity, then, which depends upon and varies with another quantity may be called a function of it, and either may be considered a function of the other.
[404] Lagrange, *Leçons sur le Calcul des Fonctions*, 1806, p. 4.
Given the quantities, we want the function of which they are the values. Simple inspection of the numbers cannot as a general rule disclose the function. In an earlier chapter (p. 124) I put before the reader certain numbers, and requested him to point out the law which they obey, and the same question will have to be asked in every case of quantitative induction. There are perhaps three methods, more or less distinct, by which we may hope to obtain an answer:
(1) By purely haphazard trial.
(2) By noting the general character of the variation of the quantities, and trying by preference functions which give a similar form of variation.
(3) By deducing from previous knowledge the form of the function which is most likely to suit.
Having numerical results we are always at liberty to invent any kind of mathematical formula we like, and then try whether, by the suitable selection of values for the unknown constant quantities, we can make it give the required results. If ever we fall upon a formula which does so, to a fair degree of approximation, there is a presumption in favour of its being the true function, although there is no certainty whatever in the matter. In this way I discovered a simple mathematical law which closely agreed with the results of my experiments on muscular exertion. This law was afterwards shown by Professor Haughton to be the true rational law according to his theory of muscular action.[405]
[405] Haughton, *Principles of Animal Mechanics*, 1873, pp. 444–450. Jevons, *Nature*, 30th of June, 1870, vol. ii. p. 158. See also the experiments of Professor Nipher, of Washington University, St. Louis, in *American Journal of Science*, vol. ix. p. 130, vol. x. p. 1; *Nature*, vol. xi. pp. 256, 276.
But the chance of succeeding in this manner is small. The number of possible functions is infinite, and even the number of comparatively simple functions is so large that the probability of falling upon the correct one by mere chance is very slight. Even when we obtain the law it is by a deductive process, not by showing that the numbers give the law, but that the law gives the numbers.
In the second way, we may, by a survey of the numbers, gain a general notion of the kind of law they are likely to obey, and we may be much assisted in this process by drawing them out in the form of a curve. We can in this way ascertain with some probability whether the curve is likely to return into itself, or whether it has infinite branches; whether such branches are asymptotic, that is, approach infinitely towards straight lines; whether it is logarithmic in character, or trigonometric. This indeed we can only do if we remember the results of previous investigations. The process is still inversely deductive, and consists in noting what laws give particular curves, and then inferring inversely that such curves belong to such laws. If we can in this way discover the class of functions to which the required law belongs, our chances of success are much increased, because our haphazard trials are now reduced within a narrower sphere. But, unless we have almost the whole curve before us, the identification of its character must be a matter of great uncertainty; and if, as in most physical investigations, we have a mere fragment of the curve, the assistance given would be quite illusory. Curves of almost any character can be made to approximate to each other for a limited extent, so that it is only by a kind of *divination* that we fall upon the actual function, unless we have theoretical knowledge of the kind of function applicable to the case.
When we have once obtained what we believe to be the correct form of function, the remainder of the work is mere mathematical computation to be performed infallibly according to fixed rules,[406] which include those employed in the determination of empirical formulæ (p. 487). The function will involve two or three or more unknown constants, the values of which we need to determine by our experimental results. Selecting some of our results widely apart and nearly equidistant, we form by means of them as many equations as there are constant quantities to be determined. The solution of these equations will then give us the constants required, and having now the actual function we can try whether it gives with sufficient accuracy the remainder of our experimental results. If not, we must either make a new selection of results to give a new set of equations, and thus obtain a new set of values for the constants, or we must acknowledge that our form of function has been wrongly chosen. If it appears that the form of function has been correctly ascertained, we may regard the constants as only approximately accurate and may proceed by the Method of Least Squares (p. 393) to determine the most probable values as given by the whole of the experimental results.
[406] Jamin, *Cours de Physique*, vol. ii. p. 50.
In most cases we shall find ourselves obliged to fall back upon the third mode, that is, anticipation of the form of the law to be expected on the ground of previous knowledge. Theory and analogical reasoning must be our guides. The general nature of the phenomenon will often indicate the kind of law to be looked for. If one form of energy or one kind of substance is being converted into another, we may expect the law of direct simple proportion. In one distinct class of cases the effect already produced influences the amount of the ensuing effect, as for instance in the cooling of a heated body, when the law will be of an exponential form. When the direction of a force influences its action, trigonometrical functions enter. Any influence which spreads freely through tridimensional space will be subject to the law of the inverse square of the distance. From such considerations we may sometimes arrive deductively and analogically at the general nature of the mathematical law required.
*The Graphical Method.*
In endeavouring to discover the mathematical law obeyed by experimental results it is often desirable to call in the aid of space-representations. Every equation involving two variable quantities corresponds to some kind of plane curve, and every plane curve may be represented symbolically in an equation containing two unknown quantities. Now in an experimental research we obtain a number of values of the variant corresponding to an equal number of values of the variable; but all the numbers are affected by more or less error, and the values of the variable will often be irregularly disposed. Even if the numbers were absolutely correct and disposed at regular intervals, there is, as we have seen, no direct mode of discovering the law, but the difficulty of discovery is much increased by the uncertainty and irregularity of the results.
Under such circumstances, the best mode of proceeding is to prepare a paper divided into equal rectangular spaces, a convenient size for the spaces being one-tenth of an inch square. The values of the variable being marked off on the lowest horizontal line, a point is marked for each corresponding value of the variant perpendicularly above that of the variable, and at such a height as corresponds to the value of the variant.
The exact scale of the drawing is not of much importance, but it may require to be adjusted according to circumstances, and different values must often be attributed to the upright and horizontal divisions, so as to make the variations conspicuous but not excessive. If a curved line be drawn through all the points or ends of the ordinates, it will probably exhibit irregular inflections, owing to the errors which affect the numbers. But, when the results are numerous, it becomes apparent which results are more divergent than others, and guided by a so-called *sense of continuity*, it is possible to trace a line among the points which will approximate to the true law more nearly than the points themselves. The accompanying figure sufficiently explains itself.
Perkins employed this graphical method with much care in exhibiting the results of his experiments on the compression of water.[407] The numerical results were marked upon a sheet of paper very exactly ruled at intervals of one-tenth of an inch, and the original marks were left in order that the reader might judge of the correctness of the curve drawn, or choose another for himself. Regnault carried the method to perfection by laying off the points with a screw dividing engine;[408] and he then formed a table of results by drawing a continuous curve, and measuring its height for equidistant values of the variable. Not only does a curve drawn in this manner enable us to infer numerical results more free from accidental errors than any of the numbers obtained directly from experiment, but the form of the curve sometimes indicates the class of functions to which our results belong.
[407] *Philosophical Transactions*, 1826, p. 544.
[408] Jamin, *Cours de Physique*, vol. ii. p. 24, &c.
Engraved sheets of paper prepared for the drawing of curves may be obtained from Mr. Stanford at Charing Cross, Messrs. W. and A. K. Johnston, of London and Edinburgh, Waterlow and Sons, Letts and Co., and probably other publishers. When we do not require great accuracy, paper ruled by the common machine-ruler into equal squares of about one-fifth or one-sixth of an inch square will serve well enough. I have met with engineers’ and surveyors’ memorandum books ruled with one-twelfth inch squares. When a number of curves have to be drawn, I have found it best to rule a good sheet of drawing paper with lines carefully adjusted at the most convenient distances, and then to prick the points of the curve through it upon another sheet fixed underneath. In this way we obtain an accurate curve upon a blank sheet, and need only introduce such division lines as are requisite to the understanding of the curve.
In some cases our numerical results will correspond, not to the height of single ordinates, but to the area of the curve between two ordinates, or the average height of ordinates between certain limits. If we measure, for instance, the quantities of heat absorbed by water when raised in temperature from 0° to 5°, from 5° to 10°, and so on, these quantities will really be represented by *areas* of the curve denoting the specific heat of water; and since the specific heat varies continuously between every two points of temperature, we shall not get the correct curve by simply laying off the quantities of heat at the mean temperatures, namely 2-1/2°, and 7-1/2°, and so on. Lord Rayleigh has shown that if we have drawn such an incorrect curve, we can with little trouble correct it by a simple geometrical process, and obtain to a close approximation the true ordinates instead of those denoting areas.[409]
[409] J. W. Strutt, *On a correction sometimes required in curves professing to represent the connexion between two physical magnitudes*. Philosophical Magazine, 4th Series, vol. xlii. p. 441.
*Interpolation and Extrapolation.*
When we have by experiment obtained two or more numerical results, and endeavour, without further experiment, to calculate intermediate results, we are said to *interpolate*. If we wish to assign by reasoning results lying beyond the limits of experiment, we may be said, using an expression of Sir George Airy, to *extrapolate*. These two operations are the same in principle, but differ in practicability. It is a matter of great scientific importance to apprehend precisely how far we can practise interpolation or extrapolation, and on what grounds we proceed.
In the first place, if the interpolation is to be more than empirical, we must have not only the experimental results, but the laws which they obey--we must in fact go through the complete process of scientific investigation. Having discovered the laws of nature applying to the case, and verified them by showing that they agree with the experiments in question, we are then in a position to anticipate the results of similar experiments. Our knowledge even now is not certain, because we cannot completely prove the truth of any assumed law, and we cannot possibly exhaust all the circumstances which may affect the result. At the best then our interpolations will partake of the want of certainty and precision attaching to all our knowledge of nature. Yet, having the supposed laws, our results will be as sure and accurate as any we can attain to. But such a complete procedure is more than we commonly mean by interpolation, which usually denotes some method of estimating in a merely approximate manner the results which might have been expected independently of a theoretical investigation.
Regarded in this light, interpolation is in reality an indeterminate problem. From given values of a function it is impossible to determine that function; for we can invent an infinite number of functions which will give those values if we are not restricted by any conditions, just as through a given series of points we can draw an infinite number of curves, if we may diverge between or beyond the points into bends and cusps as we think fit.[410] In interpolation we must in fact be guided more or less by *à priori* considerations; we must know, for instance, whether or not periodical fluctuations are to be expected. Supposing that the phenomenon is non-periodic, we proceed to assume that the function can be expressed in a limited series of the powers of the variable. The number of powers which can be included depends upon the number of experimental results available, and must be at least one less than this number. By processes of calculation, which have been already alluded to in the section on empirical formulæ, we then calculate the coefficients of the powers, and obtain an empirical formula which will give the required intermediate results. In reality, then, we return to the methods treated under the head of approximation and empirical formulæ; and interpolation, as commonly understood, consists in assuming that a curve of simple character is to pass through certain determined points. If we have, for instance, two experimental results, and only two, we assume that the curve is a straight line; for the parabolas which can be passed through two points are infinitely various in magnitude, and quite indeterminate. One straight line alone can pass through two points, and it will have an equation of the form, *y* = *mx* + *n*, the constant quantities of which can be determined from two results. Thus, if the two values for *x*, 7 and 11, give the values for *y*, 35 and 53, the solution of two equations gives *y* = 4·5 × *x* + 3·5 as the equation, and for any other value of *x*, for instance 10, we get a value of *y*, that is 48·5. When we take a mean value of *x*, namely 9, this process yields a simple mean result, namely 44. Three experimental results being given, we assume that they fall upon a portion of a parabola and algebraic calculation gives the position of any intermediate point upon the parabola. Concerning the process of interpolation as practised in the science of meteorology the reader will find some directions in the French edition of Kaëmtz’s Meteorology.[411]
[410] Herschel: Lacroix’ *Differential Calculus*, p. 551.
[411] *Cours complet de Météorologie*, Note A, p. 449.
When we have, either by direct experiment or by the use of a curve, a series of values of the variant for equidistant values of the variable, it is instructive to take the differences between each value of the variant and the next, and then the differences between those differences, and so on. If any series of differences approaches closely to zero it is an indication that the numbers may be correctly represented by a finite empirical formula; if the *n*th differences are zero, then the formula will contain only the first *n* - 1 powers of the variable. Indeed we may sometimes obtain by the calculus of differences a correct empirical formula; for if *p* be the first term of the series of values, and Δ*p*, Δ^{2}*p*, Δ^{3}*p*, be the first number in each column of differences, then the *m*th term of the series of values will be
*p* + *m*Δ*p* + *m*[(*m* - 1)/2]Δ^{2}*p* + *m*[(*m* - 1)/2][(*m* - 2)/3]Δ^{3}*p* + &c.
A closely equivalent but more practicable formula for interpolation by differences, as devised by Lagrange, will be found in Thomson and Tait’s *Elements of Natural Philosophy*, p. 115.
If no column of differences shows any tendency to become zero throughout, it is an indication that the law is of a more complicated, for instance of an exponential character, so that it requires different treatment. Dr. J. Hopkinson has suggested a method of arithmetical interpolation,[412] which is intended to avoid much that is arbitrary in the graphical method. His process will yield the same results in all hands.
[412] *On the Calculation of Empirical Formulæ. The Messenger of Mathematics*, New Series, No. 17, 1872.
So far as we can infer the results likely to be obtained by variations beyond the limits of experiment, we must proceed upon the same principles. If possible we must detect the exact laws in action, and then trust to them as a guide when we have no experience. If not, an empirical formula of the same character as those employed in interpolation is our only resource. But to extend our inference far beyond the limits of experience is exceedingly unsafe. Our knowledge is at the best only approximate, and takes no account of small tendencies. Now it usually happens that tendencies small within our limits of observation become perceptible or great under extreme circumstances. When the variable in our empirical formula is small, we are justified in overlooking the higher powers, and taking only two or three lower powers. But as the variable increases, the higher powers gain in importance, and in time yield the principal part of the value of the function.
This is no mere theoretical inference. Excepting the few primary laws of nature, such as the law of gravity, of the conservation of energy, &c., there is hardly any natural law which we can trust in circumstances widely different from those with which we are practically acquainted. From the expansion or contraction, fusion or vaporisation of substances by heat at the surface of the earth, we can form a most imperfect notion of what would happen near the centre of the earth, where the pressure almost infinitely exceeds anything possible in our experiments. The physics of the earth give us a feeble, and probably a misleading, notion of a body like the sun, in which an inconceivably high temperature is united with an inconceivably high pressure. If there are in the realms of space nebulæ consisting of incandescent and unoxidised vapours of metals and other elements, so highly heated perhaps that chemical composition is out of the question, we are hardly able to treat them as subjects of scientific inference. Hence arises the great importance of experiments in which we investigate the properties of substances under extreme circumstances of cold or heat, density or rarity, intense electric excitation, &c. This insecurity in extending our inferences arises from the approximate character of our measurements. Had we the power of appreciating infinitely small quantities, we should by the principle of continuity discover some trace of every change which a substance could undergo under unattainable circumstances. By observing, for instance, the tension of aqueous vapour between 0° and 100° C., we ought theoretically to be able to infer its tension at every other temperature; but this is out of the question practically because we cannot really ascertain the law precisely between those temperatures.
Many instances might be given to show that laws which appear to represent correctly the results of experiments within certain limits altogether fail beyond those limits. The experiments of Roscoe and Dittmar, on the absorption of gases in water[413] afford interesting illustrations, especially in the case of hydrochloric acid, the quantity of which dissolved in water under different pressures follows very closely a linear law of variation, from which however it diverges widely at low pressures.[414] Herschel, having deduced from observations of the double star γ Virginis an elliptic orbit for the motion of one component round the centre of gravity of both, found that for a time the motion of the star agreed very well with this orbit. Nevertheless divergence began to appear and after a time became so great that an entirely new orbit, of more than double the dimensions of the old one, had ultimately to be adopted.[415]
[413] Watts’ *Dictionary of Chemistry*, vol. ii. p. 790.
[414] *Quarterly Journal of the Chemical Society*, vol. viii. p. 15.
[415] *Results of Observations at the Cape of Good Hope*, p. 293.
*Illustrations of Empirical Quantitative Laws.*
Although our object in quantitative inquiry is to discover the exact or rational formulæ, expressing the laws which apply to the subject, it is instructive to observe in how many important branches of science, no precise laws have yet been detected. The tension of aqueous vapour at different temperatures has been determined by a succession of eminent experimentalists--Dalton, Kaëmtz, Dulong, Arago, Magnus, and Regnault--and by the last mentioned the measurements were conducted with extraordinary care. Yet no incontestable general law has been established. Several functions have been proposed to express the elastic force of the vapour as depending on the temperature. The first form is that of Young, namely F = (*a* + *b t*)^{m}, in which *a*, *b*, and *m* are unknown quantities to be determined by observation. Roche proposed, on theoretical grounds, a complicated formula of an exponential form, and a third form of function is that of Biot,[416] as follows--log F = *a* + *b*α^{t} + *c*β^{t}. I mention these formulæ, because they well illustrate the feeble powers of empirical inquiry. None of the formulæ can be made to correspond closely with experimental results, and the two last forms correspond almost equally well. There is very little probability that the real law has been reached, and it is unlikely that it will be discovered except by deduction from mechanical theory.
[416] Jamin, *Cours de Physique*, vol. ii. p. 138.
Much ingenious labour has been spent upon the discovery of some general law of atmospheric refraction. Tycho Brahe and Kepler commenced the inquiry: Cassini first formed a table of refractions, calculated on theoretical grounds: Newton entered into some profound investigations upon the subject: Brooke Taylor, Bouguer, Simpson, Bradley, Mayer, and Kramp successively attacked the question, which is of the highest practical importance as regards the correction of astronomical observations. Laplace next laboured on the subject without exhausting it, and Brinkley and Ivory have also treated it. The true law is yet undiscovered. A closely connected problem, that regarding the relation between the pressure and elevation in different strata of the atmosphere, has received the attention of a long succession of physicists and was most carefully investigated by Laplace. Yet no invariable and general law has been detected. The same may be said concerning the law of human mortality; abundant statistics on this subject are available, and many hypotheses more or less satisfactory have been put forward as to the form of the curve of mortality, but it seems to be impossible to discover more than an approximate law.
It may perhaps be urged that in such subjects no single invariable law can be expected. The atmosphere may be divided into several variable strata which by their unconnected changes frustrate the exact calculations of astronomers. Human life may be subject at different ages to a succession of different influences incapable of reduction under any one law. The results observed may in fact be aggregates of an immense number of separate results each governed by its own separate laws, so that the subjects may be complicated beyond the possibility of complete resolution by empirical methods. This is certainly true of the mathematical functions which must some time or other be introduced into the science of political economy.
*Simple Proportional Variation.*
When we first treat numerical results in any novel kind of investigation, our impression will probably be that one quantity varies in *simple proportion* to another, so as to obey the law *y* = *mx* + *n*. We must learn to distinguish carefully between the cases where this proportionality is really, and where it is only apparently true. In considering the principles of approximation we found that a small portion of any curve will appear to be a straight line. When our modes of measurement are comparatively rude, we must expect to be unable to detect the curvature. Kepler made meritorious attempts to discover the law of refraction, and he approximated to it when he observed that the angles of incidence and refraction *if small* bear a constant ratio to each other. Angles when small are nearly as their sines, so that he reached an approximate result of the true law. Cardan assumed, probably as a mere guess, that the force required to sustain a body on an inclined plane was simply proportional to the angle of elevation of the plane. This is approximately the case when the angle is small, but in reality the law is much more complicated, the power required being proportional to the sine of the angle. The early thermometer-makers were unaware whether the expansion of mercury was proportional or not to the heat communicated to it, and it is only in the present century that we have learnt it to be not so. We now know that even gases obey the law of uniform expansion by heat only in an approximate manner. Until reason to the contrary is shown, we should do well to look upon every law of simple proportion as only provisionally true.
Nevertheless many important laws of nature are in the form of simple proportions. Wherever a cause acts in independence of its previous effects, we may expect this relation. An accelerating force acts equally upon a moving and a motionless body. Hence the velocity produced is in simple proportion to the force, and to the duration of its uniform action. As gravitating bodies never interfere with each other’s gravity, this force is in direct simple proportion to the mass of each of the attracting bodies, the mass being measured by, or proportional to inertia. Similarly, in all cases of “direct unimpeded action,” as Herschel has remarked,[417] we may expect simple proportion to manifest itself. In such cases the equation expressing the relation may have the simple form *y* = *mx*.
[417] *Preliminary Discourse*, &c., p. 152.Return to passage
A similar relation holds true when there is conversion of one substance or form of energy into another. The quantity of a compound is equal to the quantity of the elements which combine. The heat produced in friction is exactly proportional to the mechanical energy absorbed. It was experimentally proved by Faraday that “the chemical power of the current of electricity is in direct proportion to the quantity of electricity which passes.” When an electric current is produced, the quantity of electric energy is simply proportional to the weight of metal dissolved. If electricity is turned into heat, there is again simple proportion. Wherever, in fact, one thing is but another thing with a new aspect, we may expect to find the law of simple proportion. But it is only in the most elementary cases that this simple relation will hold true. Simple conditions do not, generally speaking, produce simple results. The planets move in approximate circles round the sun, but the apparent motions, as seen from the earth, are very various. All those motions, again, are summed up in the law of gravity, of no great complexity; yet men never have been, and never will be, able to exhaust the complications of action and reaction arising from that law, even among a small number of planets. We should be on our guard against a tendency to assume that the connection of cause and effect is one of direct proportion. Bacon reminds us of the woman in Æsop’s fable, who expected that her hen, with a double measure of barley, would lay two eggs a day instead of one, whereas it grew fat, and ceased to lay any eggs at all. It is a wise maxim that the half is often better than the whole.
XXIII. THE USE OF HYPOTHESIS
If the views upheld in this work be correct, all inductive investigation consists in the marriage of hypothesis and experiment. When facts are in our possession, we frame an hypothesis to explain their relations, and by the success of this explanation is the value of the hypothesis to be judged. In the invention and treatment of such hypotheses, we must avail ourselves of the whole body of science already accumulated, and when once we have obtained a probable hypothesis, we must not rest until we have verified it by comparison with new facts. We must endeavour by deductive reasoning to anticipate such phenomena, especially those of a singular and exceptional nature, as would happen if the hypothesis be true. Out of the infinite number of experiments which are possible, theory must lead us to select those critical ones which are suitable for confirming or negativing our anticipations.
This work of inductive investigation cannot be guided by any system of precise and infallible rules, like those of deductive reasoning. There is, in fact, nothing to which we can apply rules of method, because the laws of nature must be in our possession before we can treat them. If there were any rule of inductive method, it would direct us to make an exhaustive arrangement of facts in all possible orders. Given the specimens in a museum, we might arrive at the best classification by going systematically through all possible classifications, and, were we endowed with infinite time and patience, this would be an effective method. It is the method by which the first simple steps are taken in an incipient branch of science. Before the dignified name of science is applicable, some coincidences will force themselves upon the attention. Before there was a science of meteorology observant persons learned to associate clearness of the atmosphere with coming rain, and a colourless sunset with fine weather. Knowledge of this kind is called *empirical*, as seeming to come directly from experience; and there is a considerable portion of knowledge which bears this character.
We may be obliged to trust to the casual detection of coincidences in those branches of knowledge where we are deprived of the aid of any guiding notions; but a little reflection will show the utter insufficiency of haphazard experiment, when applied to investigations of a complicated nature. At the best, it will be the simple identity, or partial identity, of classes, as illustrated in pages 127 or 134, which can be thus detected. It was pointed out that, even when a law of nature involves only two circumstances, and there are one hundred distinct circumstances which may possibly be connected, there will be no less than 4,950 pairs of circumstances between which coincidence may exist. When a law involves three or more circumstances, the possible number of relations becomes vastly greater. When considering the subject of combinations and permutations, it became apparent that we could never cope with the possible variety of nature. An exhaustive examination of the possible metallic alloys, or chemical compounds, was found to be out of the question (p. 191).
It is on such considerations that we can explain the very small additions made to our knowledge by the alchemists. Many of them were men of the greatest acuteness, and their indefatigable labours were pursued through many centuries. A few things were discovered by them, but a true insight into nature, now enables chemists to discover more useful facts in a year than were yielded by the alchemists during many centuries. There can be no doubt that Newton was an alchemist, and that he often laboured night and day at alchemical experiments. But in trying to discover the secret by which gross metals might be rendered noble, his lofty powers of deductive investigation were wholly useless. Deprived of all guiding clues, his experiments were like those of all the alchemists, purely tentative and haphazard. While his hypothetical and deductive investigations have given us the true system of the Universe, and opened the way in almost all the great branches of natural philosophy, the whole results of his tentative experiments are comprehended in a few happy guesses, given in his celebrated “Queries.”
Even when we are engaged in apparently passive observation of a phenomenon, which we cannot modify experimentally, it is advantageous that our attention should be guided by theoretical anticipations. A phenomenon which seems simple is, in all probability, really complex, and unless the mind is actively engaged in looking for particular details, it is likely that the critical circumstances will be passed over. Bessel regretted that no distinct theory of the constitution of comets had guided his observations of Halley’s comet;[418] in attempting to verify or refute a hypothesis, not only would there be a chance of establishing a true theory, but if confuted, the confutation would involve a store of useful observations.
[418] Tyndall, *On Cometary Theory*, Philosophical Magazine, April 1869. 4th Series, vol. xxxvii. p. 243.
It would be an interesting work, but one which I cannot undertake, to trace out the gradual reaction which has taken place in recent times against the purely empirical or Baconian theory of induction. Francis Bacon, seeing the futility of the scholastic logic, which had long been predominant, asserted that the accumulation of facts and the orderly abstraction of axioms, or general laws from them, constituted the true method of induction. Even Bacon was not wholly unaware of the value of hypothetical anticipation. In one or two places he incidentally acknowledges it, as when he remarks that the subtlety of nature surpasses that of reason, adding that “axioms abstracted from particular facts in a careful and orderly manner, readily suggest and mark out new particulars.”
Nevertheless Bacon’s method, as far as we can gather the meaning of the main portions of his writings, would correspond to the process of empirically collecting facts and exhaustively classifying them, to which I alluded. The value of this method may be estimated historically by the fact that it has not been followed by any of the great masters of science. Whether we look to Galileo, who preceded Bacon, to Gilbert, his contemporary, or to Newton and Descartes, Leibnitz and Huyghens, his successors, we find that discovery was achieved by the opposite method to that advocated by Bacon. Throughout Newton’s works, as I shall show, we find deductive reasoning wholly predominant, and experiments are employed, as they should be, to confirm or refute hypothetical anticipations of nature. In my “Elementary Lessons in Logic” (p. 258), I stated my belief that there was no kind of reference to Bacon in Newton’s works. I have since found that Newton does once or twice employ the expression *experimentum crucis* in his “Opticks,” but this is the only expression, so far as I am aware, which could indicate on the part of Newton direct or indirect acquaintance with Bacon’s writings.[419]
[419] See *Philosophical Transactions*, abridged by Lowthorp. 4th edit. vol. i. p. 130. I find that opinions similar to those in the text have been briefly expressed by De Morgan in his remarkable preface to *From Matter to Spirit*, by C.D., pp. xxi. xxii.
Other great physicists of the same age were equally prone to the use of hypotheses rather than the blind accumulation of facts in the Baconian manner. Hooke emphatically asserts in his posthumous work on Philosophical Method, that the first requisite of the Natural Philosopher is readiness at guessing the solution of phenomena and making queries. “He ought to be very well skilled in those several kinds of philosophy already known, to understand their several hypotheses, suppositions, collections, observations, &c., their various ways of ratiocinations and proceedings, the several failings and defects, both in their way of raising and in their way of managing their several theories: for by this means the mind will be somewhat more ready at guessing at the solution of many phenomena almost at first sight, and thereby be much more prompt at making queries, and at tracing the subtlety of Nature, and in discovering and searching into the true reason of things.”
We find Horrocks, again, than whom no one was more filled with the scientific spirit, telling us how he tried theory after theory in order to discover one which was in accordance with the motions of Mars.[420] Huyghens, who possessed one of the most perfect philosophical intellects, followed the deductive process combined with continual appeal to experiment, with a skill closely analogous to that of Newton. As to Descartes and Leibnitz, they fell into excess in the use of hypothesis, since they sometimes adopted hypothetical reasoning to the exclusion of experimental verification. Throughout the eighteenth century science was supposed to be advancing by the pursuance of the Baconian method, but in reality hypothetical investigation was the main instrument of progress. It is only in the present century that physicists began to recognise this truth. So much opprobrium had been attached by Bacon to the use of hypotheses, that we find Young speaking of them in an apologetic tone. “The practice of advancing general principles and applying them to particular instances is so far from being fatal to truth in all sciences, that when those principles are advanced on sufficient grounds, it constitutes the essence of true philosophy;”[421] and he quotes cases in which Davy trusted to his theories rather than his experiments.
[420] Horrocks, *Opera Posthuma* (1673), p. 276.
[421] Young’s *Works*, vol. i. p. 593.
Herschel, who was both a practical physicist and an abstract logician, entertained the deepest respect for Bacon, and made the “Novum Organum” as far as possible the basis of his own admirable *Discourse on the Study of Natural Philosophy*. Yet we find him in Chapter VII. recognising the part which the formation and verification of theories takes in the higher and more general investigations of physical science. J. S. Mill carried on the reaction by describing the Deductive Method in which ratiocination, that is deductive reasoning, is employed for the discovery of new opportunities of testing and verifying an hypothesis. Nevertheless throughout the other parts of his system he inveighed against the value of the deductive process, and even asserted that empirical inference from particulars to particulars is the true type of reasoning. The irony of fate will probably decide that the most original and valuable part of Mill’s System of Logic is irreconcilable with those views of the syllogism and of the nature of inference which occupy the main part of the treatise, and are said to have effected a revolution in logical science. Mill would have been saved from much confusion of thought had he not failed to observe that the inverse use of deduction constitutes induction. In later years Professor Huxley has strongly insisted upon the value of hypothesis. When he advocates the use of “working hypotheses” he means no doubt that any hypothesis is better that none, and that we cannot avoid being guided in our observations by some hypothesis or other. Professor Tyndall’s views as to the use of the Imagination in the pursuit of Science put the same truth in another light.
It ought to be pointed out that Neil in his *Art of Reasoning*, a popular but able exposition of the principles of Logic, published in 1853, fully recognises in Chapter XI. the value and position of hypothesis in the discovery of truth. He endeavours to show, too (p. 109), that Francis Bacon did not object to the use of hypothesis.
The true course of inductive procedure is that which has yielded all the more lofty results of science. It consists in *Anticipating Nature*, in the sense of forming hypotheses as to the laws which are probably in operation; and then observing whether the combinations of phenomena are such as would follow from the laws supposed. The investigator begins with facts and ends with them. He uses facts to suggest probable hypotheses; deducing other facts which would happen if a particular hypothesis is true, he proceeds to test the truth of his notion by fresh observations. If any result prove different from what he expects, it leads him to modify or to abandon his hypothesis; but every new fact may give some new suggestion as to the laws in action. Even if the result in any case agrees with his anticipations, he does not regard it as finally confirmatory of his theory, but proceeds to test the truth of the theory by new deductions and new trials.
In such a process the investigator is assisted by the whole body of science previously accumulated. He may employ analogy, as I shall point out, to guide him in the choice of hypotheses. The manifold connections between one science and another give him clues to the kind of laws to be expected, and out of the infinite number of possible hypotheses he selects those which are, as far as can be foreseen at the moment, most probable. Each experiment, therefore, which he performs is that most likely to throw light upon his subject, and even if it frustrate his first views, it tends to put him in possession of the correct clue.
*Requisites of a good Hypothesis.*
There is little difficulty in pointing out to what condition an hypothesis must conform in order to be accepted as probable and valid. That condition, as I conceive, is the single one of enabling us to infer the existence of phenomena which occur in our experience. *Agreement with fact is the sole and sufficient test of a true hypothesis.*
Hobbes has named two conditions which he considers requisite in an hypothesis, namely (1) That it should be conceivable and not absurd; (2) That it should allow of phenomena being necessarily inferred. Boyle, in noticing Hobbes’ views, proposed to add a third condition, to the effect that the hypothesis should not be inconsistent with any other truth on phenomenon of nature.[422] I think that of these three conditions, the first cannot be accepted, unless by *inconceivable* and *absurd* we mean self-contradictory or inconsistent with the laws of thought and nature. I shall have to point out that some satisfactory theories involve suppositions which are wholly *inconceivable* in a certain sense of the word, because the mind cannot sufficiently extend its ideas to frame a notion of the actions supposed to take place. That the force of gravity should act instantaneously between the most distant parts of the planetary system, or that a ray of violet light should consist of about 700 billions of vibrations in a second, are statements of an inconceivable and absurd character in one sense; but they are so far from being opposed to fact that we cannot on any other suppositions account for phenomena observed. But if an hypothesis involve self-contradiction, or is inconsistent with known laws of nature, it is self-condemned. We cannot even apply deductive reasoning to a self-contradictory notion; and being opposed to the most general and certain laws known to us, the primary laws of thought, it thereby conspicuously fails to agree with facts. Since nature, again, is never self-contradictory, we cannot at the same time accept two theories which lead to contradictory results. If the one agrees with nature, the other cannot. Hence if there be a law which we believe with high probability to be verified by observation, we must not frame an hypothesis in conflict with it, otherwise the hypothesis will necessarily be in disagreement with observation. Since no law or hypothesis is proved, indeed, with absolute certainty, there is always a chance, however slight, that the new hypothesis may displace the old one; but the greater the probability which we assign to that old hypothesis, the greater must be the evidence required in favour of the new and conflicting one.
[422] Boyle’s *Physical Examen*, p. 84.
I assert, then, that there is but one test of a good hypothesis, namely, *its conformity with observed facts*; but this condition may be said to involve three constituent conditions, nearly equivalent to those suggested by Hobbes and Boyle, namely:--
(1) That it allow of the application of deductive reasoning and the inference of consequences capable of comparison with the results of observation.
(2) That it do not conflict with any laws of nature, or of mind, which we hold to be true.
(3) That the consequences inferred do agree with facts of observation.
*Possibility of Deductive Reasoning.*
As the truth of an hypothesis is to be proved by its conformity with fact, the first condition is that we be able to apply methods of deductive reasoning, and learn what should happen according to such an hypothesis. Even if we could imagine an object acting according to laws hitherto wholly unknown it would be useless to do so, because we could never decide whether it existed or not. We can only infer what would happen under supposed conditions by applying the knowledge of nature we possess to those conditions. Hence, as Boscovich truly said, we are to understand by hypotheses “not fictions altogether arbitrary, but suppositions conformable to experience or analogy.” It follows that every hypothesis worthy of consideration must suggest some likeness, analogy, or common law, acting in two or more things. If, in order to explain certain facts, *a*, *a′*, *a″*, &c., we invent a cause A, then we must in some degree appeal to experience as to the mode in which A will act. As the laws of nature are not known to the mind intuitively, we must point out some other cause, B, which supplies the requisite notions, and all we do is to invent a fourth term to an analogy. As B is to its effects *b*, *b′*, *b″*, &c., so is A to its effects *a*, *a′*, *a″*, &c. When we attempt to explain the passage of light and heat radiations through space unoccupied by matter, we imagine the existence of the so-called *ether*. But if this ether were wholly different from anything else known to us, we should in vain try to reason about it. We must apply to it at least the laws of motion, that is we must so far liken it to matter. And as, when applying those laws to the elastic medium air, we are able to infer the phenomena of sound, so by arguing in a similar manner concerning ether we are able to infer the existence of light phenomena corresponding to what do occur. All that we do is to take an elastic substance, increase its elasticity immensely, and denude it of gravity and some other properties of matter, but we must retain sufficient likeness to matter to allow of deductive calculations.
The force of gravity is in some respects an incomprehensible existence, but in other respects entirely conformable to experience. We observe that the force is proportional to mass, and that it acts in entire independence of other matter which may be present or intervening. The law of the decrease of intensity, as the square of the distance increases, is observed to hold true of light, sound, and other influences emanating from a point, and spreading uniformly through space. The law is doubtless connected with the properties of space, and is so far in agreement with our necessary ideas.
It may be said, however, that no hypothesis can be so much as framed in the mind unless it be more or less conformable to experience. As the material of our ideas is derived from sensation we cannot figure to ourselves any agent, but as endowed with some of the properties of matter. All that the mind can do in the creation of new existences is to alter combinations, or the intensity of sensuous properties. The phenomenon of motion is familiar to sight and touch, and different degrees of rapidity are also familiar; we can pass beyond the limits of sense, and imagine the existence of rapid motion, such as our senses could not observe. We know what is elasticity, and we can therefore in a way figure to ourselves elasticity a thousand or a million times greater than any which is sensuously known to us. The waves of the ocean are many times higher than our own bodies; other waves, are many times less; continue the proportion, and we ultimately arrive at waves as small as those of light. Thus it is that the powers of mind enable us from a sensuous basis to reason concerning agents and phenomena different in an unlimited degree. If no hypothesis then can be absolutely opposed to sense, accordance with experience must always be a question of degree.
In order that an hypothesis may allow of satisfactory comparison with experience, it must possess definiteness and in many cases mathematical exactness allowing of the precise calculation of results. We must be able to ascertain whether it does or does not agree with facts. The theory of vortices is an instance to the contrary, for it did not present any mode of calculating the exact relations between the distances and periods of the planets and satellites; it could not, therefore, undergo that rigorous testing to which Newton scrupulously submitted his theory of gravity before its promulgation. Vagueness and incapability of precise proof or disproof often enable a false theory to live; but with those who love truth, vagueness should excite suspicion. The upholders of the ancient doctrine of Nature’s abhorrence of a vacuum, had been unable to anticipate the important fact that water would not rise more than 33 feet in a common suction pump. Nor when the fact was pointed out could they explain it, except by introducing a special alteration of the theory to the effect that Nature’s abhorrence of a vacuum was limited to 33 feet.
*Consistency with the Laws of Nature.*
In the second place an hypothesis must not be contradictory to what we believe to be true concerning Nature. It must not involve self-inconsistency which is opposed to the highest and simplest laws, namely, those of Logic. Neither ought it to be irreconcilable with the simple laws of motion, of gravity, of the conservation of energy, nor any parts of physical science which we consider to be established beyond reasonable doubt. Not that we are absolutely forbidden to entertain such an hypothesis, but if we do so we must be prepared to disprove some of the best demonstrated truths in the possession of mankind. The fact that conflict exists means that the consequences of the theory are not verified if previous discoveries are correct, and we must therefore show that previous discoveries are incorrect before we can verify our theory.
An hypothesis will be exceedingly improbable, not to say absurd, if it supposes a substance to act in a manner unknown in other cases; for it then fails to be verified in our knowledge of that substance. Several physicists, especially Euler and Grove, have supposed that we might dispense with an ethereal basis of light, and infer from the interstellar passage of rays that there was a kind of rare gas occupying space. But if so, that gas must be excessively rare, as we may infer from the apparent absence of an atmosphere around the moon, and from other facts known to us concerning gases and the atmosphere; yet it must possess an elastic force at least a billion times as great as atmospheric air at the earth’s surface, in order to account for the extreme rapidity of light rays. Such an hypothesis then is inconsistent with our knowledge concerning gases.
Provided that there be no clear and absolute conflict with known laws of nature, there is no hypothesis so improbable or apparently inconceivable that it may not be rendered probable, or even approximately certain, by a sufficient number of concordances. In fact the two best founded and most successful theories in physical science involve the most absurd suppositions. Gravity is a force which appears to act between bodies through vacuous space; it is in positive contradiction to the old dictum that nothing can act but through some medium. It is even more puzzling that the force acts in perfect indifference to intervening obstacles. Light in spite of its extreme velocity shows much respect to matter, for it is almost instantaneously stopped by opaque substances, and to a considerable extent absorbed and deflected by transparent ones. But to gravity all media are, as it were, absolutely transparent, nay non-existent; and two particles at opposite points of the earth affect each other exactly as if the globe were not between. The action is, so far as we can observe, instantaneous, so that every particle of the universe is at every moment in separate cognisance, as it were, of the relative position of every other particle throughout the universe at that same moment of time. Compared with such incomprehensible conditions, the theory of vortices deals with commonplace realities. Newton’s celebrated saying *hypotheses non fingo*, bears the appearance of irony; and it was not without apparent grounds that Leibnitz and the continental philosophers charged Newton with re-introducing occult powers and qualities.
The undulatory theory of light presents almost equal difficulties of conception. We are asked by physical philosophers to give up our prepossessions, and to believe that interstellar space which seems empty is not empty at all, but filled with *something* immensely more solid and elastic than steel. As Young himself remarked,[423] “the luminiferous ether, pervading all space, and penetrating almost all substances, is not only highly elastic, but absolutely solid!!!” Herschel calculated the force which may be supposed, according to the undulatory theory of light, to be constantly exerted at each point in space, and finds it to be 1,148,000,000,000 times the elastic force of ordinary air at the earth’s surface, so that the pressure of ether per square inch must be about seventeen billions of pounds.[424] Yet we live and move without appreciable resistance through this medium, immensely harder and more elastic than adamant. All our ordinary notions must be laid aside in contemplating such an hypothesis; yet it is no more than the observed phenomena of light and heat force us to accept. We cannot deny even the strange suggestion of Young, that there may be independent worlds, some possibly existing in different parts of space, but others perhaps pervading each other unseen and unknown in the same space.[425] For if we are bound to admit the conception of this adamantine firmament, it is equally easy to admit a plurality of such. We see, then, that mere difficulties of conception must not discredit a theory which otherwise agrees with facts, and we must only reject hypotheses which are inconceivable in the sense of breaking distinctly the primary laws of thought and nature.
[423] Young’s *Works*, vol. i. p. 415.
[424] *Familiar Lectures on Scientific Subjects*, p. 282.
[425] Young’s *Works*, vol. i. p. 417.
*Conformity with Facts.*
Before we accept a new hypothesis it must be shown to agree not only with the previously known laws of nature, but also with the particular facts which it is framed to explain. Assuming that these facts are properly established, it must agree with all of them. A single absolute conflict between fact and hypothesis, is fatal to the hypothesis; *falsa in uno, falsa in omnibus*.
Seldom, indeed, shall we have a theory free from difficulties and apparent inconsistency with facts. Though one real inconsistency would overturn the most plausible theory, yet there is usually some probability that the fact may be misinterpreted, or that some supposed law of nature, on which we are relying, may not be true. It may be expected, moreover, that a good hypothesis, besides agreeing with facts already noticed, will furnish us with distinct credentials by enabling us to anticipate deductively series of facts which are not already connected and accounted for by any equally probable hypothesis. We cannot lay down any precise rule as to the number of accordances which can establish the truth of an hypothesis, because the accordances will vary much in value. While, on the one hand, no finite number of accordances will give entire certainty, the probability of the hypothesis will increase very rapidly with the number of accordances. Almost every problem in science thus takes the form of a balance of probabilities. It is only when difficulty after difficulty has been successfully explained away, and decisive *experimenta crucis* have, time after time, resulted in favour of our theory, that we can venture to assert the falsity of all objections.
The sole real test of an hypothesis is its accordance with fact. Descartes’ celebrated system of vortices is exploded, not because it was intrinsically absurd and inconceivable, but because it could not give results in accordance with the actual motions of the heavenly bodies. The difficulties of conception involved in the apparatus of vortices, are child’s play compared with those of gravitation and the undulatory theory already described. Vortices are on the whole plausible suppositions; for planets and satellites bear at first sight much resemblance to objects carried round in whirlpools, an analogy which doubtless suggested the theory. The failure was in the first and third requisites; for, as already remarked, the theory did not allow of precise calculation of planetary motions, and was thus incapable of rigorous verification. But so far as we can institute a comparison, facts are entirely against the vortices. Newton did not ridicule the theory as absurd, but showed[426] that it was “pressed with many difficulties.” He carefully pointed out that the Cartesian theory was inconsistent with the laws of Kepler, and would represent the planets as moving more rapidly at their aphelia than at their perihelia.[427] The rotatory motion of the sun and planets on their own axes is in striking conflict with the revolutions of the satellites carried round them; and comets, the most flimsy of bodies, calmly pursue their courses in elliptic paths, irrespective of the vortices which they pass through. We may now also point to the interlacing orbits of the minor planets as a new and insuperable difficulty in the way of the Cartesian ideas.
[426] *Principia*, bk. iii. Prop. 43. General Scholium.
[427] Ibid. bk. ii. Sect. ix. Prop. 53.
Newton, though he established the best of theories, was also capable of proposing one of the worst; and if we want an instance of a theory decisively contradicted by facts, we have only to turn to his views concerning the origin of natural colours. Having analysed, with incomparable skill, the origin of the colours of thin plates, he suggests that the colours of all bodies are determined in like manner by the size of their ultimate particles. A thin plate of a definite thickness will reflect a definite colour; hence, if broken up into fragments it will form a powder of the same colour. But, if this be a sufficient explanation of coloured substances, then every coloured fluid ought to reflect the complementary colour of that which it transmits. Colourless transparency arises, according to Newton, from particles being too minute to reflect light; but if so, every black substance should be transparent. Newton himself so acutely felt this last difficulty as to suggest that true blackness is due to some internal refraction of the rays to and fro, and an ultimate stifling of them, which he did not attempt to explain further. Unless some other process comes into operation, neither refraction nor reflection, however often repeated, will destroy the energy of light. The theory therefore gives no account, as Brewster shows, of 24 parts out of 25 of the light which falls upon a black coal, and the remaining part which is reflected from the lustrous surface is equally inconsistent with the theory, because fine coal-dust is almost entirely devoid of reflective power.[428] It is now generally believed that the colours of natural bodies are due to the unequal absorption of rays of light of different refrangibility.
[428] Brewster’s *Life of Newton*, 1st edit. chap. vii.
*Experimentum Crucis.*
As we deduce more and more conclusions from a theory, and find them verified by trial, the probability of the theory increases in a rapid manner; but we never escape the risk of error altogether. Absolute certainty is beyond the powers of inductive investigation, and the most plausible supposition may ultimately be proved false. Such is the groundwork of similarity in nature, that two very different conditions may often give closely similar results. We sometimes find ourselves therefore in possession of two or more hypotheses which both agree with so many experimental facts as to have great appearance of truth. Under such circumstances we have need of some new experiment, which shall give results agreeing with one hypothesis but not with the other.
Any such experiment which decides between two rival theories may be called an *Experimentum Crucis*, an Experiment of the Finger Post. Whenever the mind stands, as it were, at cross-roads and knows not which way to select, it needs some decisive guide, and Bacon therefore assigned great importance and authority to instances which serve in this capacity. The name given by Bacon has become familiar; it is almost the only one of Bacon’s figurative expressions which has passed into common use. Even Newton, as I have mentioned (p. 507), used the name.
I do not think, indeed, that the common use of the word at all agrees with that intended by Bacon. Herschel says that “we make an experiment of the crucial kind when we form combinations, and put in action causes from which some particular one shall be deliberately excluded, and some other purposely admitted.”[429] This, however, seems to be the description of any special experiment not made at haphazard. Pascal’s experiment of causing a barometer to be carried to the top of the Puy-de-Dôme has often been considered as a perfect *experimentum crucis*, if not the first distinct one on record;[430] but if so, we must dignify the doctrine of Nature’s abhorrence of a vacuum with the position of a rival theory. A crucial experiment must not simply confirm one theory, but must negative another; it must decide a mind which is in equilibrium, as Bacon says,[431] between two equally plausible views. “When in search of any nature, the understanding comes to an equilibrium, as it were, or stands suspended as to which of two or more natures the cause of nature inquired after should be attributed or assigned, by reason of the frequent and common occurrence of several natures, then these Crucial Instances show the true and inviolable association of one of these natures to the nature sought, and the uncertain and separable alliance of the other, whereby the question is decided, the former nature admitted for the cause, and the other rejected. These instances, therefore, afford great light, and have a kind of overruling authority, so that the course of interpretation will sometimes terminate in them, or be finished by them.”
[429] *Discourse on the Study of Natural Philosophy*, p. 151.
[430] Ibid. p. 229.
[431] *Novum Organum*, bk. ii. Aphorism 36.
The long-continued strife between the Corpuscular and Undulatory theories of light forms the best possible illustration of an Experimentum Crucis. It is remarkable in how plausible a manner both these theories agreed with the ordinary laws of geometrical optics, relating to reflection and refraction. According to the first law of motion a moving particle proceeds in a perfectly straight line, when undisturbed by extraneous forces. If the particle being perfectly elastic, strike a perfectly elastic plane, it will bound off in such a path that the angles of incidence and reflection will be equal. Now a ray of light proceeds in a straight line, or appears to do so, until it meets a reflecting body, when its path is altered in a manner exactly similar to that of the elastic particle. Here is a remarkable correspondence which probably suggested to Newton’s mind the hypothesis that light consists of minute elastic particles moving with excessive rapidity in straight lines. The correspondence was found to extend also to the law of simple refraction; for if particles of light be supposed capable of attracting matter, and being attracted by it at insensibly small distances, then a ray of light, falling on the surface of a transparent medium, will suffer an increase in its velocity perpendicular to the surface, and the law of sines is the consequence. This remarkable explanation of the law of refraction had doubtless a very strong effect in leading Newton to entertain the corpuscular theory, and he appears to have thought that the analogy between the propagation of rays of light and the motion of bodies was perfectly exact, whatever might be the actual nature of light.[432] It is highly remarkable, again, that Newton was able to give by his corpuscular theory, a plausible explanation of the inflection of light as discovered by Grimaldi. The theory would indeed have been a very probable one could Newton’s own law of gravity have applied; but this was out of the question, because the particles of light, in order that they may move in straight lines, must be devoid of any influence upon each other.
[432] *Principia*, bk. i. Sect. xiv. Prop. 96. Scholium. *Opticks*, Prop. vi. 3rd edit. p. 70.
The Huyghenian or Undulatory theory of light was also able to explain the same phenomena, but with one remarkable difference. If the undulatory theory be true, light must move more slowly in a dense refracting medium than in a rarer one; but the Newtonian theory assumed that the attraction of the dense medium caused the particles of light to move more rapidly than in the rare medium. On this point, then, there was complete discrepancy between the theories, and observation was required to show which theory was to be preferred. Now by simply cutting a uniform plate of glass into two pieces, and slightly inclining one piece so as to increase the length of the path of a ray passing through it, experimenters were able to show that light does move more slowly in glass than in air.[433] More recently Fizeau and Foucault independently measured the velocity of light in air and in water, and found that the velocity is greater in air.[434]
[433] Airy’s *Mathematical Tracts*, 3rd edit. pp. 286–288.
[434] Jamin, *Cours de Physique*, vol. iii. p. 372.
There are a number of other points at which experience decides against Newton, and in favour of Huyghens and Young. Laplace pointed out that the attraction supposed to exist between matter and the corpuscular particles of light would cause the velocity of light to vary with the size of the emitting body, so that if a star were 250 times as great in diameter as our sun, its attraction would prevent the emanation of light altogether.[435] But experience shows that the velocity of light is uniform, and independent of the magnitude of the emitting body, as it should be according to the undulatory theory. Lastly, Newton’s explanation of diffraction or inflection fringes of colours was only *plausible*, and not true; for Fresnel ascertained that the dimensions of the fringes are not what they would be according to Newton’s theory.
[435] Young’s *Lectures on Natural Philosophy* (1845), vol. i. p. 361.
Although the Science of Light presents us with the most beautiful examples of crucial experiments and observations, instances are not wanting in other branches of science. Copernicus asserted, in opposition to the ancient Ptolemaic theory, that the earth moved round the sun, and he predicted that if ever the sense of sight could be rendered sufficiently acute and powerful, we should see phases in Mercury and Venus. Galileo with his telescope was able, in 1610 to verify the prediction as regards Venus, and subsequent observations of Mercury led to a like conclusion. The discovery of the aberration of light added a new proof, still further strengthened by the more recent determination of the parallax of fixed stars. Hooke proposed to prove the existence of the earth’s diurnal motion by observing the deviation of a falling body, an experiment successfully accomplished by Benzenberg; and Foucault’s pendulum has since furnished an additional indication of the same motion, which is indeed also apparent in the trade winds. All these are crucial facts in favour of the Copernican theory.
*Descriptive Hypotheses.*
There are hypotheses which we may call *descriptive hypotheses*, and which serve for little else than to furnish convenient names. When a phenomenon is of an unusual kind, we cannot even speak of it without using some analogy. Every word implies some resemblance between the thing to which it is applied, and some other thing, which fixes the meaning of the word. If we are to speak of what constitutes electricity, we must search for the nearest analogy, and as electricity is characterised by the rapidity and facility of its movements, the notion of a fluid of a very subtle character presents itself as appropriate. There is the single-fluid and the double-fluid theory of electricity, and a great deal of discussion has been uselessly spent upon them. The fact is, that if these theories be understood as more than convenient modes of describing the phenomena, they are altogether invalid. The analogy extends only to the rapidity of motion, or rather the fact that a phenomenon occurs successively at different points of the body. The so-called electric fluid adds nothing to the weight of the conductor, and to suppose that it really consists of particles of matter is even more absurd than to reinstate the corpuscular theory of light. A far closer analogy exists between electricity and light undulations, which are about equally rapid in propagation. We shall probably continue for a long time to talk of the *electric fluid*, but there can be no doubt that this expression represents merely a phase of molecular motion, a wave of disturbance. The invalidity of these fluid theories is shown moreover in the fact that they have not led to the invention of a single new experiment.
Among these merely descriptive hypotheses I should place Newton’s theory of Fits of Easy Reflection and Refraction. That theory did not do more than describe what took place. It involved no analogy to other phenomena of nature, for Newton could not point to any other substance which went through these extraordinary fits. We now know that the true analogy would have been waves of sound, of which Newton had acquired in other respects so complete a comprehension. But though the notion of interference of waves had distinctly occurred to Hooke, Newton failed to see how the periodic phenomena of light could be connected with the periodic character of waves. His hypothesis fell because it was out of analogy with everything else in nature, and it therefore did not allow him, as in other cases, to descend by mathematical deduction to consequences which could be verified or refuted.
We are at freedom to imagine the existence of a new agent, and to give it an appropriate name, provided there are phenomena incapable of explanation from known causes. We may speak of *vital force* as occasioning life, provided that we do not take it to be more than a name for an undefined something giving rise to inexplicable facts, just as the French chemists called Iodine the Substance X, so long as they were unaware of its real character and place in chemistry.[436] Encke was quite justified in speaking of the *resisting medium* in space so long as the retardation of his comet could not be otherwise accounted for. But such hypotheses will do much harm whenever they divert us from attempts to reconcile the facts with known laws, or when they lead us to mix up discrete things. Because we speak of vital force we must not assume that it is a really existing physical force like electricity; we do not know what it is. We have no right to confuse Encke’s supposed resisting medium with the basis of light without distinct evidence of identity. The name protoplasm, now so familiarly used by physiologists, is doubtless legitimate so long as we do not mix up different substances under it, or imagine that the name gives us any knowledge of the obscure origin of life. To name a substance protoplasm no more explains the infinite variety of forms of life which spring out of the substance, than does the *vital force* which may be supposed to reside in the protoplasm. Both expressions are mere names for an inexplicable series of causes which out of apparently similar conditions produce the most diverse results.
[436] Paris, *Life of Davy*, p. 274.Return to passage
Hardly to be distinguished from descriptive hypotheses are certain imaginary objects which we frame for the ready comprehension of a subject. The mathematician, in treating abstract questions of probability, finds it convenient to represent the conditions by a concrete hypothesis in the shape of a ballot-box. Poisson proved the principle of the inverse method of probabilities by imagining a number of ballot-boxes to have their contents mixed in one great ballot-box (p. 244). Many such devices are used by mathematicians. The Ptolemaic theory of *cycles* and *epi-cycles* was no grotesque and useless work of the imagination, but a perfectly valid mode of analysing the motions of the heavenly bodies; in reality it is used by mathematicians at the present day. Newton employed the pendulum as a means of representing the nature of an undulation. Centres of gravity, oscillation, &c., poles of the magnet, lines of force, are other imaginary existences employed to assist our thoughts (p. 364). Such devices may be called *Representative Hypotheses*, and they are only permissible so far as they embody analogies. Their further consideration belongs either to the subject of Analogy, or to that of language and representation, founded upon analogy.