CHAPTER XXVII.

GENERALISATION.

I have endeavoured to show in preceding chapters that all inductive reasoning is an inverse application of deductive reasoning, and consists in demonstrating that the consequences of certain assumed laws agree with facts of nature gathered by active or passive observation. The fundamental process of reasoning, as stated in the outset, consists in inferring of a thing what we know of similar objects, and it is on this principle that the whole of deductive reasoning, whether simply logical or mathematico-logical, is founded. All inductive reasoning must be founded on the same principle. It might seem that by a plain use of this principle we could avoid the complicated processes of induction and deduction, and argue directly from one particular case to another, as Mill proposed. If the Earth, Venus, Mars, Jupiter, and other planets move in elliptic orbits, cannot we dispense with elaborate precautions, and assert that Neptune, Ceres, and the last discovered planet must do so likewise? Do we not know that Mr. Gladstone must die, because he is like other men? May we not argue that because some men die therefore he must? Is it requisite to ascend by induction to the general proposition “all men must die,” and then descend by deduction from that general proposition to the case of Mr. Gladstone? My answer undoubtedly is that we must ascend to general propositions. The fundamental principle of the substitution of similars gives us no warrant in affirming of Mr. Gladstone what we know of other men, because we cannot be sure that Mr. Gladstone is exactly similar to other men. Until his death we cannot be perfectly sure that he possesses all the attributes of other men; it is a question of probability, and I have endeavoured to explain the mode in which the theory of probability is applied to calculate the probability that from a series of similar events we may infer the recurrence of like events under identical circumstances. There is then no such process as that of inferring from particulars to particulars. A careful analysis of the conditions under which such an inference appears to be made, shows that the process is really a general one, and that what is inferred of a particular case might be inferred of all similar cases. All reasoning is essentially general, and all science implies generalisation. In the very birth-time of philosophy this was held to be so: “Nulla scientia est de individuis, sed de solis universalibus,” was the doctrine of Plato, delivered by Porphyry. And Aristotle[489] held a like opinion--Οὐδεμία δὲ τέχνη σκοπεȋ τὸ καθ’ ἕκαστον ... τὸ δὲ καθ’ ἕκαστον ἄπειρον καὶ οὐκ ἐπιστητόν. “No art treats of particular cases; for particulars are infinite and cannot be known.” No one who holds the doctrine that reasoning may be from particulars to particulars, can be supposed to have the most rudimentary notion of what constitutes reasoning and scíence.

[489] Aristotle’s *Rhetoric*, Liber I. 2. 11.

At the same time there can be no doubt that practically what we find to be true of many similar objects will probably be true of the next similar object. This is the result to which an analysis of the Inverse Method of Probabilities leads us, and, in the absence of precise data from which we may calculate probabilities, we are usually obliged to make a rough assumption that similars in some respects are similars in other respects. Thus it comes to pass that a large part of the reasoning processes in which scientific men are engaged, consists in detecting similarities between objects, and then rudely assuming that the like similarities will be detected in other cases.

*Distinction of Generalisation and Analogy.*

There is no distinction but that of degree between what is known as reasoning by *generalisation* and reasoning by *analogy*. In both cases from certain observed resemblances we infer, with more or less probability, the existence of other resemblances. In generalisation the resemblances have great extension and usually little intension, whereas in analogy we rely upon the great intension, the extension being of small amount (p. 26). If we find that the qualities A and B are associated together in a great many instances, and have never been found separate, it is highly probable that on the next occasion when we meet with A, B will also be present, and *vice versâ*. Thus wherever we meet with an object possessing gravity, it is found to possess inertia also, nor have we met with any material objects possessing inertia without discovering that they also possess gravity. The probability has therefore become very great, as indicated by the rules founded on the Inverse Method of Probabilities (p. 257), that whenever in the future we meet an object possessing either of the properties of gravity and inertia, it will be found on examination to possess the other of these properties. This is a clear instance of the employment of generalisation.

In analogy, on the other hand, we reason from likeness in many points to likeness in other points. The qualities or points of resemblance are now numerous, not the objects. At the poles of Mars are two white spots which resemble in many respects the white regions of ice and snow at the poles of the earth. There probably exist no other similar objects with which to compare these, yet the exactness of the resemblance enables us to infer, with high probability, that the spots on Mars consist of ice and snow. In short, many points of resemblance imply many more. From the appearance and behaviour of those white spots we infer that they have all the chemical and physical properties of frozen water. The inference is of course only probable, and based upon the improbability that aggregates of many qualities should be formed in a like manner in two or more cases, without being due to some uniform condition or cause.

In reasoning by analogy, then, we observe that two objects ABCDE.... and A′B′C′D′E′.... have many like qualities, as indicated by the identity of the letters, and we infer that, since the first has another quality, X, we shall discover this quality in the second case by sufficiently close examination. As Laplace says,--“Analogy is founded on the probability that similar things have causes of the same kind, and produce the same effects. The more perfect this similarity, the greater is this probability.”[490] The nature of analogical inference is aptly described in the work on Logic attributed to Kant, where the rule of ordinary induction is stated in the words, “*Eines in vielen, also in allen*,” one quality in many things, therefore in all; and the rule of analogy is “*Vieles in einem, also auch das übrige in demselben*,”[491] many (qualities) in one, therefore also the remainder in the same. It is evident that there may be intermediate cases in which, from the identity of a moderate number of objects in several properties, we may infer to other objects. Probability must rest either upon the number of instances or the depth of resemblance, or upon the occurrence of both in sufficient degrees. What there is wanting in extension must be made up by intension, and *vice versâ*.

[490] *Essai Philosophique sur les Probabilités*, p. 86.

[491] Kant’s *Logik*, § 84, Königsberg, 1800, p. 207.

*Two Meanings of Generalisation.*

The term generalisation, as commonly used, includes two processes which are of different character, but are often closely associated together. In the first place, we generalise when we recognise even in two objects a common nature. We cannot detect the slightest similarity without opening the way to inference from one case to the other. If we compare a cubical crystal with a regular octahedron, there is little apparent similarity; but, as soon as we perceive that either can be produced by the symmetrical modification of the other, we discover a groundwork of similarity in the crystals, which enables us to infer many things of one, because they are true of the other. Our knowledge of ozone took its rise from the time when the similarity of smell, attending electric sparks, strokes of lightning, and the slow combustion of phosphorus, was noticed by Schönbein. There was a time when the rainbow was an inexplicable phenomenon--a portent, like a comet, and a cause of superstitious hopes and fears. But we find the true spirit of science in Roger Bacon, who desires us to consider the objects which present the same colours as the rainbow; he mentions hexagonal crystals from Ireland and India, but he bids us not suppose that the hexagonal form is essential, for similar colours may be detected in many transparent stones. Drops of water scattered by the oar in the sun, the spray from a water-wheel, the dewdrops lying on the grass in the summer morning, all display a similar phenomenon. No sooner have we grouped together these apparently diverse instances, than we have begun to generalise, and have acquired a power of applying to one instance what we can detect of others. Even when we do not apply the knowledge gained to new objects, our comprehension of those already observed is greatly strengthened and deepened by learning to view them as particular cases of a more general property.

A second process, to which the name of generalisation is often given, consists in passing from a fact or partial law to a multitude of unexamined cases, which we believe to be subject to the same conditions. Instead of merely recognising similarity as it is brought before us, we predict its existence before our senses can detect it, so that generalisation of this kind endows us with a prophetic power of more or less probability. Having observed that many substances assume, like water and mercury, the three states of solid, liquid, and gas, and having assured ourselves by frequent trial that the greater the means we possess of heating and cooling, the more substances we can vaporise and freeze, we pass confidently in advance of fact, and assume that all substances are capable of these three forms. Such a generalisation was accepted by Lavoisier and Laplace before many of the corroborative facts now in our possession were known. The reduction of a single comet beneath the sway of gravity was considered sufficient indication that all comets obey the same power. Few persons doubted that the law of gravity extended over the whole heavens; certainly the fact that a few stars out of many millions manifest the action of gravity, is now held to be sufficient evidence of its general extension over the visible universe.

*Value of Generalisation.*

It might seem that if we know particular facts, there can be little use in connecting them together by a general law. The particulars must be more full of useful information than an abstract general statement. If we know, for instance, the properties of an ellipse, a circle, a parabola, and hyperbola, what is the use of learning all these properties over again in the general theory of curves of the second degree? If we understand the phenomena of sound and light and water-waves separately, what is the need of erecting a general theory of waves, which, after all, is inapplicable to practice until resolved again into particular cases? But, in reality, we never do obtain an adequate knowledge of particulars until we regard them as cases of the general. Not only is there a singular delight in discovering the many in the one, and the one in the many, but there is a constant interchange of light and knowledge. Properties which are unapparent in the hyperbola may be readily observed in the ellipse. Most of the complex relations which old geometers discovered in the circle will be reproduced *mutatis mutandis* in the other conic sections. The undulatory theory of light might have been unknown at the present day, had not the theory of sound supplied hints by analogy. The study of light has made known many phenomena of interference and polarisation, the existence of which had hardly been suspected in the case of sound, but which may now be sought out, and perhaps found to possess unexpected interest. The careful study of water-waves shows how waves alter in form and velocity with varying depth of water. Analogous changes may some time be detected in sound waves. Thus there is mutual interchange of aid.

“Every study of a generalisation or extension,” De Morgan has well said,[492] “gives additional power over the particular form by which the generalisation is suggested. Nobody who has ever returned to quadratic equations after the study of equations of all degrees, or who has done the like, will deny my assertion that οὐ βλέπει βλέπων may be predicated of any one who studies a branch or a case, without afterwards making it part of a larger whole. Accordingly it is always worth while to generalise, were it only to give power over the *particular*. This principle, of daily familiarity to the mathematician, is almost unknown to the logician.”

[492] *Syllabus of a Proposed System of Logic*, p. 34.

*Comparative Generality of Properties.*

Much of the value of science depends upon the knowledge which we gradually acquire of the different degrees of generality of properties and phenomena of various kinds. The use of science consists in enabling us to act with confidence, because we can foresee the result. Now this foresight must rest upon the knowledge of the powers which will come into play. That knowledge, indeed, can never be certain, because it rests upon imperfect induction, and the most confident beliefs and predictions of the physicist may be falsified. Nevertheless, if we always estimate the probability of each belief according to the due teaching of the data, and bear in mind that probability when forming our anticipations, we shall ensure the minimum of disappointment. Even when he cannot exactly apply the theory of probabilities, the physicist may acquire the habit of making judgments in general agreement with its principles and results.

Such is the constitution of nature, that the physicist learns to distinguish those properties which have wide and uniform extension, from those which vary between case and case. Not only are certain laws distinctly laid down, with their extension carefully defined, but a scientific training gives a kind of tact in judging how far other laws are likely to apply under any particular circumstances. We learn by degrees that crystals exhibit phenomena depending upon the directions of the axes of elasticity, which we must not expect in uniform solids. Liquids, compared even with non-crystalline solids, exhibit laws of far less complexity and variety; and gases assume, in many respects, an aspect of nearly complete uniformity. To trace out the branches of science in which varying degrees of generality prevail, would be an inquiry of great interest and importance; but want of space, if there were no other reason, would forbid me to attempt it, except in a very slight manner.

Gases, so far as they are really gaseous, not only have exactly the same properties in all directions of space, but one gas exactly resembles other gases in many qualities. All gases expand by heat, according to the same law, and by nearly the same amount; the specific heats of equivalent weights are equal, and the densities are exactly proportional to the atomic weights. All such gases obey the general law, that the volume multiplied by the pressure, and divided by the absolute temperature, is constant or nearly so. The laws of diffusion and transpiration are the same in all cases, and, generally speaking, all physical laws, as distinguished from chemical laws, apply equally to all gases. Even when gases differ in chemical or physical properties, the differences are minor in degree. Thus the differences of viscosity are far less marked than in the liquid and solid states. Nearly all gases, again, are colourless, the exceptions being chlorine, the vapours of iodine, bromine, and a few other substances.

Only in one single point, so far as I am aware, do gases present distinguishing marks unknown or nearly so, in the solid and liquid states. I mean as regards the light given off when incandescent. Each gas when sufficiently heated, yields its own peculiar series of rays, arising from the free vibrations of the constituent parts of the molecules. Hence the possibility of distinguishing gases by the spectroscope. But the molecules of solids and liquids appear to be continually in conflict with each other, so that only a confused *noise* of atoms is produced, instead of a definite series of luminous chords. At the same temperature, accordingly, all solids and liquids give off nearly the same rays when strongly heated, and we have in this case an exception to the greater generality of properties in gases.

Liquids are in many ways intermediate in character between gases and solids. While incapable of possessing different elasticity in different directions, and thus denuded of the rich geometrical complexity of solids, they retain the variety of density, colour degrees of transparency, great diversity in surface tension, viscosity, coefficients of expansion, compressibility, and many other properties which we observe in solids, but not for the most part in gases. Though our knowledge of the physical properties of liquids is much wanting in generality at present, there is ground to hope that by degrees laws connecting and explaining the variations may be traced out.

Solids are in every way contrasted to gases. Each solid substance has its own peculiar degree of density, hardness, compressibility, transparency, tenacity, elasticity, power of conducting heat and electricity, magnetic properties, capability of producing frictional electricity, and so forth. Even different specimens of the same kind of substance will differ widely, according to the accidental treatment received. And not only has each substance its own specific properties, but, when crystallised, its properties vary in each direction with regard to the axes of crystallisation. The velocity of radiation, the rate of conduction of heat, the coefficients of expansibility and compressibility, the thermo-electric properties, all vary in different crystallographic directions.

It is probable that many apparent differences between liquids, and even between solids, will be explained when we learn to regard them under exactly corresponding circumstances. The extreme generality of the properties of gases is in reality only true at an infinitely high temperature, when they are all equally remote from their condensing points. Now, it is found that if we compare liquids--for instance, different kinds of alcohols--not at equal temperatures, but at points equally distant from their respective boiling points, the laws and coefficients of expansion are nearly equal. The vapour-tensions of liquids also are more nearly equal, when compared at corresponding points, and the boiling-points appear in many cases to be simply related to the chemical composition. No doubt the progress of investigation will enable us to discover generality, where at present we only see variety and puzzling complexity.

In some cases substances exhibit the same physical properties in the liquid as in the solid state. Lead has a high refractive power, whether in solution, or in solid salts, crystallised or vitreous. The magnetic power of iron is conspicuous, whatever be its chemical condition; indeed, the magnetic properties of substances, though varying with temperature, seem not to be greatly affected by other physical changes. Colour, absorptive power for heat or light rays, and a few other properties are also often the same in liquids and gases. Iodine and bromine possess a deep colour whenever they are chemically uncombined. Nevertheless, we can seldom argue safely from the properties of a substance in one condition to those in another condition. Ice is an insulator, water a conductor of electricity, and the same contrast exists in most other substances. The conducting power of a liquid for electricity increases with the temperature, while that of a solid decreases. By degrees we may learn to distinguish between those properties of matter which depend upon the intimate construction of the chemical molecule, and those which depend upon the contact, conflict, mutual attraction, or other relations of distinct molecules. The properties of a substance with respect to light seem generally to depend upon the molecule; thus, the power of certain substances to cause the plane of polarisation of a ray of light to rotate, is exactly the same whatever be its degree of density, or the diluteness of the solution in which it is contained. Taken as a whole, the physical properties of substances and their quantitative laws, present a problem of infinite complexity, and centuries must elapse before any moderately complete generalisations on the subject become possible.

*Uniform Properties of all Matter.*

Some laws are held to be true of all matter in the universe absolutely, without exception, no instance to the contrary having ever been noticed. This is the case with the laws of motion, as laid down by Galileo and Newton. It is also conspicuously true of the law of universal gravitation. The rise of modern physical science may perhaps be considered as beginning at the time when Galileo showed, in opposition to the Aristotelians, that matter is equally affected by gravity, irrespective of its form, magnitude, or texture. All objects fall with equal rapidity, when disturbing causes, such as the resistance of the air, are removed or allowed for. That which was rudely demonstrated by Galileo from the leaning tower of Pisa, was proved by Newton to a high degree of approximation, in an experiment which has been mentioned (p. 443).

Newton formed two pendulums, as nearly as possible the same in outward shape and size by taking two equal round wooden boxes, and suspending them by equal threads, eleven feet long. The pendulums were therefore equally subject to the resistance of the air. He filled one box with wood, and in the centre of oscillation of the other he placed an equal weight of gold. The pendulums were then equal in weight as well as in size; and, on setting them simultaneously in motion, Newton found that they vibrated for a length of time with equal vibrations. He tried the same experiment with silver, lead, glass, sand, common salt, water, and wheat, in place of the gold, and ascertained that the motion of his pendulum was exactly the same whatever was the kind of matter inside.[493] He considered that a difference of a thousandth part would have been apparent. The reader must observe that the pendulums were made of equal weight only in order that they might suffer equal retardation from the air. The meaning of the experiment is that all substances manifest exactly equal acceleration from the force of gravity, and that therefore the inertia or resistance of matter to force, which is the only independent measure of mass known to us, is always proportional to gravity.

[493] *Principia*, bk. iii. Prop. VI. Motte’s translation, vol. ii. p. 220.

These experiments of Newton were considered conclusive up to very recent times, when certain discordances between the theory and observations of the movements of planets led Nicolai, in 1826, to suggest that the equal gravitation of different kinds of matter might not be absolutely exact. It is perfectly philosophical thus to call in question, from time to time, some of the best accepted laws. On this occasion Bessel carefully repeated the experiments of Newton with pendulums composed of ivory, glass, marble, quartz, meteoric stones, &c., but was unable to detect the least difference. This conclusion is also confirmed by the ultimate agreement of all the calculations of physical astronomy based upon it. Whether the mass of Jupiter be calculated from the motion of its own satellites, from the effect upon the small planets, Vesta, Juno, &c., or from the perturbation of Encke’s Comet, the results are closely accordant, showing that precisely the same law of gravity applies to the most different bodies which we can observe. The gravity of a body, again, appears to be entirely independent of its other physical conditions, being totally unaffected by any alteration in the temperature, density, electric or magnetic condition, or other physical properties of the substance.

One paradoxical result of the law of equal gravitation is the theorem of Torricelli, to the effect that all liquids of whatever density fall or flow with equal rapidity. If there be two equal cisterns respectively filled with mercury and water, the mercury, though thirteen times as heavy, would flow from an aperture neither more rapidly nor more slowly than the water, and the same would be true of ether, alcohol, and other liquids, allowance being made, however, for the resistance of the air, and the differing viscosities of the liquids.

In its exact equality and its perfect independence of all circumstances, except mass and distance, the force of gravity stands apart from all the other forces and phenomena of nature, and has not yet been brought into any relation with them except through the general principle of the conservation of energy. Magnetic attraction, as remarked by Newton, follows very different laws, depending upon the chemical quality and molecular structure of each particular substance.

We must remember that in saying “all matter gravitates,” we exclude from the term matter the basis of light-undulations, which is immensely more extensive in amount, and obeys in many respects the laws of mechanics. This adamantine substance appears, so far as can be ascertained, to be perfectly uniform in its properties when existing in space unoccupied by matter. Light and heat are conveyed by it with equal velocity in all directions, and in all parts of space so far as observation informs us. But the presence of gravitating matter modifies the density and mechanical properties of the so-called ether in a way which is yet quite unexplained.[494]

Leaving gravity, it is somewhat difficult to discover other laws which are equally true of all matter. Boerhaave was considered to have established that all bodies expand by heat; but not only is the expansion very different in different substances, but we now know positive exceptions. Many liquids and a few solids contract by heat at certain temperatures. There are indeed other relations of heat to matter which seem to be universal and uniform; all substances begin to give off rays of light at the same temperature, according to the law of Draper; and gases will not be an exception if sufficiently condensed, as in the experiments of Frankland. Grove considers it to be universally true that all bodies in combining produce heat; with the doubtful exception of sulphur and selenium, all solids in becoming liquids, and all liquids in becoming gases, absorb heat; but the quantities of heat absorbed vary with the chemical qualities of the matter. Carnot’s Thermodynamic Law is held to be exactly true of all matter without distinction; it expresses the fact that the amount of mechanical energy which might be theoretically obtained from a certain amount of heat energy depends only upon the change of the temperatures, so that whether an engine be worked by water, air, alcohol, ammonia, or any other substance, the result would theoretically be the same, if the boiler and condenser were maintained at similar temperatures.

[494] Professor Lovering has pointed out how obscure and uncertain the ideas of scientific men about this ether are, in his interesting Presidential Address before the American Association at Hartford, 1874. *Silliman’s Journal*, October 1874, p. 297. *Philosophical Magazine*, vol. xlviii. p. 493.

*Variable Properties of Matter.*

I have enumerated some of the few properties of matter, which are manifested in exactly the same manner by all substances, whatever be their differences of chemical or physical constitution. But by far the greater number of qualities vary in degree; substances are more or less dense, more or less transparent, more or less compressible, more or less magnetic, and so on. One common result of the progress of science is to show that qualities supposed to be entirely absent from many substances are present only in so low a degree of intensity that the means of detection were insufficient. Newton believed that most bodies were quite unaffected by the magnet; Faraday and Tyndall have rendered it very doubtful whether any substance whatever is wholly devoid of magnetism, including under that term diamagnetism. We are rapidly learning to believe that there are no substances absolutely opaque, or non-conducting, non-electric, non-elastic, non-viscous, non-compressible, insoluble, infusible, or non-volatile. All tends to become a matter of degree, or sometimes of direction. There may be some substances oppositely affected to others, as ferro-magnetic substances are oppositely affected to diamagnetics, or as substances which contract by heat are opposed to those which expand; but the tendency is certainly for every affection of one kind of matter to be represented by something similar in other kinds. On this account one of Newton’s rules of philosophising seems to lose all validity; he said, “Those qualities of bodies which are not capable of being heightened, and remitted, and which are found in all bodies on which experiment can be made, must be considered as universal qualities of all bodies.” As far as I can see, the contrary is more probable, namely, that qualities variable in degree will be found in every substance in a greater or less degree.

It is remarkable that Newton whose method of investigation was logically perfect, seemed incapable of generalising and describing his own procedure. His celebrated “Rules of Reasoning in Philosophy,” described at the commencement of the third book of the *Principia*, are of questionable truth, and still more questionable value.

*Extreme Instances of Properties.*

Although substances usually differ only in degree, great interest may attach to particular substances which manifest a property in a conspicuous and intense manner. Every branch of physical science has usually been developed from the attention forcibly drawn to some singular substance. Just as the loadstone disclosed magnetism and amber frictional electricity, so did Iceland spar show the existence of double refraction, and sulphate of quinine the phenomenon of fluorescence. When one such startling instance has drawn the attention of the scientific world, numerous less remarkable cases of the phenomenon will be detected, and it will probably prove that the property in question is actually universal to all matter. Nevertheless, the extreme instances retain their interest, partly in a historical point of view, partly because they furnish the most convenient substances for experiment.

Francis Bacon was fully aware of the value of such examples, which he called *Ostensive Instances* or Light-giving, Free and Predominant Instances. “They are those,” he says,[495] “which show the nature under investigation naked, in an exalted condition, or in the highest degree of power; freed from impediments, or at least by its strength predominating over and suppressing them.” He mentions quicksilver as an ostensive instance of weight or density, thinking it not much less dense than gold, and more remarkable than gold as joining density to liquidity. The magnet is mentioned as an ostensive instance of attraction. It would not be easy to distinguish clearly between these ostensive instances and those which he calls *Instantiae Monodicae*, or *Irregulares*, or *Heteroclitae*, under which he places whatever is extravagant in its properties or magnitude, or exhibits least similarity to other things, such as the sun and moon among the heavenly bodies, the elephant among animals, the letter *s* among letters, or the magnet among stones.[496]

[495] *Novum Organum*, bk. ii. Aphorisms, 24, 25.

[496] Ibid. Aph. 28.

In optical science great use has been made of the high dispersive power of the transparent compounds of lead, that is, the power of giving a long spectrum (p. 432). Dollond, having noticed this peculiar dispersive power in lenses made of flint glass, employed them to produce an achromatic arrangement. The element strontium presents a contrast to lead in this respect, being characterised by a remarkably low dispersive power; but I am not aware that this property has yet been turned to account.

Compounds of lead have both a high dispersive and a high refractive index, and in the latter respect they proved very useful to Faraday. Having spent much labour in preparing various kinds of optical glass, Faraday happened to form a compound of lead, silica, and boracic acid, now known as *heavy glass*, which possessed an intensely high refracting power. Many years afterwards in attempting to discover the action of magnetism upon light he failed to detect any effect, as has been already mentioned, (p. 588), until he happened to test a piece of the heavy glass. The peculiar refractive power of this medium caused the magnetic strain to be apparent, and the rotation of the plane of polarisation was discovered.

In almost every part of physical science there is some substance of powers pre-eminent for the special purpose to which it is put. Rock-salt is invaluable for its extreme diathermancy or transparency to the least refrangible rays of the spectrum. Quartz is equally valuable for its transparency, as regards the ultra-violet or most refrangible rays. Diamond is the most highly refracting substance which is at the same time transparent; were it more abundant and easily worked it would be of great optical importance. Cinnabar is distinguished by possessing a power of rotating the plane of polarisation of light, from 15 to 17 times as much as quartz. In electric experiments copper is employed for its high conducting powers and exceedingly low magnetic properties; iron is of course indispensable for its enormous magnetic powers; while bismuth holds a like place as regards its diamagnetic powers, and was of much importance in Tyndall’s decisive researches upon the polar character of the diamagnetic force.[497] In regard to magne-crystallic action the mineral cyanite is highly remarkable, being so powerfully affected by the earth’s magnetism, that, when delicately suspended, it assumes a constant position with regard to the magnetic meridian, and may almost be used like the compass needle. Sodium is distinguished by its unique light-giving powers, which are so extraordinary that probably one half of the whole number of stars in the heavens have a yellow tinge in consequence.

[497] *Philosophical Transactions* (1856) vol. cxlvi. p. 246.

It is remarkable that water, though the most common of all fluids, is distinguished in almost every respect by extreme qualities. Of all known substances water has the highest specific heat, being thus peculiarly fitted for the purpose of warming and cooling, to which it is often put. It rises by capillary attraction to a height more than twice that of any other liquid. In the state of ice it is nearly twice as dilatable by heat as any other known solid substance.[498] In proportion to its density it has a far higher surface tension than any other substance, being surpassed in absolute tension only by mercury; and it would not be difficult to extend considerably the list of its remarkable and useful properties.

[498] *Philosophical Magazine*, 4th Series, January 1870, vol. xxxix. p. 2.

Under extreme instances we may include cases of remarkably low powers or qualities. Such cases seem to correspond to what Bacon calls *Clandestine Instances*, which exhibit a given nature in the least intensity, and as it were in a rudimentary state.[499] They may often be important, he thinks, as allowing the detection of the cause of the property by difference. I may add that in some cases they may be of use in experiments. Thus hydrogen is the least dense of all known substances, and has the least atomic weight. Liquefied nitrous oxide has the lowest refractive index of all known fluids.[500] The compounds of strontium have the lowest dispersive power. It is obvious that a property of very low degree may prove as curious and valuable a phenomenon as a property of very high degree.

[499] *Novum Organum*, bk. ii. Aphorism 25.

[500] Faraday’s *Experimental Researches in Chemistry and Physics*, p. 93.

*The Detection of Continuity.*

We should bear in mind that phenomena which are in reality of a closely similar or even identical nature, may present to the senses very different appearances. Without a careful analysis of the changes which take place, we may often be in danger of widely separating facts and processes, which are actually instances of the same law. Extreme difference of degree or magnitude is a frequent cause of error. It is truly difficult at the first moment to recognise any similarity between the gradual rusting of a piece of iron, and the rapid combustion of a heap of straw. Yet Lavoisier’s chemical theory was founded upon the similarity of the oxydising process in one case and the other. We have only to divide the iron into excessively small particles to discover that it is really the more combustible of the two, and that it actually takes fire spontaneously and burns like tinder. It is the excessive slowness of the process in the case of a massive piece of iron which disguises its real character.

If Xenophon reports truly, Socrates was misled by not making sufficient allowance for extreme differences of degree and quantity. Anaxagoras held that the sun is a fire, but Socrates rejected this opinion, on the ground that we can look at a fire, but not at the sun, and that plants grow by sunshine while they are killed by fire. He also pointed out that a stone heated in a fire is not luminous, and soon cools, whereas the sun ever remains equally luminous and hot.[501] All such mistakes evidently arise from not perceiving that difference of quantity may be so extreme as to assume the appearance of difference of quality. It is the least creditable thing we know of Socrates, that after pointing out these supposed mistakes of earlier philosophers, he advised his followers not to study astronomy.

[501] *Memorabilia*, iv. 7.

Masses of matter of very different size may be expected to exhibit apparent differences of conduct, arising from the various intensity of the forces brought into play. Many persons have thought it requisite to imagine occult forces producing the suspension of the clouds, and there have even been absurd theories representing cloud particles as minute water-balloons buoyed up by the warm air within them. But we have only to take proper account of the enormous comparative resistance which the air opposes to the fall of minute particles, to see that all cloud particles are probably constantly falling through the air, but so slowly that there is no apparent effect. Mineral matter again is always regarded as inert and incapable of spontaneous movement. We are struck by astonishment on observing in a powerful microscope, that every kind of solid matter suspended in extremely minute particles in pure water, acquires an oscillatory movement, often so marked as to resemble dancing or skipping. I conceive that this movement is due to the comparatively vast intensity of chemical action when exerted upon minute particles, the effect being 5,000 or 10,000 greater in proportion to the mass than in fragments of an inch diameter (p. 406).

Much that was formerly obscure in the science of electricity arose from the extreme differences of intensity and quantity in which this form of energy manifests itself. Between the brilliant explosive discharge of a thunder-cloud and the gentle continuous current produced by two pieces of metal and some dilute acid, there is no apparent analogy whatever. It was therefore a work of great importance when Faraday demonstrated the identity of the forces in action, showing that common frictional electricity would decompose water like that from the voltaic battery. The relation of the phenomena became plain when he succeeded in showing that it would require 800,000 discharges of his large Leyden battery to decompose one single grain of water. Lightning was now seen to be electricity of excessively high tension, but extremely small quantity, the difference being somewhat analogous to that between the force of one million gallons of water falling through one foot, and one gallon of water falling through one million feet. Faraday estimated that one grain of water acting on four grains of zinc, would yield electricity enough for a great thunderstorm.

It was long believed that electrical conductors and insulators belonged to two opposed classes of substances. Between the inconceivable rapidity with which the current passes through pure copper wire, and the apparently complete manner in which it is stopped by a thin partition of gutta-percha or gum-lac, there seemed to be no resemblance. Faraday again laboured successfully to show that these were but the extreme cases of a chain of substances varying in all degrees in their powers of conduction. Even the best conductors, such as pure copper or silver, offer resistance to the electric current. The other metals have considerably higher powers of resistance, and we pass gradually down through oxides and sulphides. The best insulators, on the other hand, allow of an atomic induction which is the necessary antecedent of conduction. Hence Faraday inferred that whether we can measure the effect or not, all substances discharge electricity more or less.[502] One consequence of this doctrine must be, that every discharge of electricity produces an induced current. In the case of the common galvanic current we can readily detect the induced current in any parallel wire or other neighbouring conductor, and can separate the opposite currents which arise at the moments when the original current begins and ends. But a discharge of high tension electricity like lightning, though it certainly occupies time and has a beginning and an end, yet lasts so minute a fraction of a second, that it would be hopeless to attempt to detect and separate the two opposite induced currents, which are nearly simultaneous and exactly neutralise each other. Thus an apparent failure of analogy is explained away, and we are furnished with another instance of a phenomenon incapable of observation and yet theoretically known to exist.[503]

[502] *Experimental Researches in Electricity*, Series xii. vol. i. p. 420.

[503] *Life of Faraday*, vol. ii. p. 7.

Perhaps the most extraordinary case of the detection of unsuspected continuity is found in the discovery of Cagniard de la Tour and Professor Andrews, that the liquid and gaseous conditions of matter are only remote points in a continuous course of change. Nothing is at first sight more apparently distinct than the physical condition of water and aqueous vapour. At the boiling-point there is an entire breach of continuity, and the gas produced is subject to laws incomparably more simple than the liquid from which it arose. But Cagniard de la Tour showed that if we maintain a liquid under sufficient pressure its boiling point may be indefinitely raised, and yet the liquid will ultimately assume the gaseous condition with but a small increase of volume. Professor Andrews, recently following out this course of inquiry, has shown that liquid carbonic acid may, at a particular temperature (30°·92 C.), and under the pressure of 74 atmospheres, be at the same time in a state indistinguishable from that of liquid and gas. At higher pressures carbonic acid may be made to pass from a palpably liquid state to a truly gaseous state without any abrupt change whatever. As the pressure is greater the abruptness of the change from liquid to gas gradually decreases, and finally vanishes. Similar phenomena or an approximation to them have been observed in other liquids, and there is little doubt that we may make a wide generalisation, and assert that, under adequate pressure, every liquid might be made to pass into a gas without breach of continuity.[504] The liquid state, moreover, is considered by Professor Andrews to be but an intermediate step between the solid and gaseous conditions. There are various indications that the process of melting is not perfectly abrupt; and could experiments be made under adequate pressures, it is believed that every solid could be made to pass by insensible degrees into the state of liquid, and subsequently into that of gas.

[504] *Nature*, vol. ii. p. 278.

These discoveries appear to open the way to most important and fundamental generalisations, but it is probable that in many other cases phenomena now regarded as discrete may be shown to be different degrees of the same process. Graham was of opinion that chemical affinity differs but in degree from the ordinary attraction which holds different particles of a body together. He found that sulphuric acid continued to evolve heat when mixed even with the fiftieth equivalent of water, so that there seemed to be no distinct limit to chemical affinity. He concludes, “There is reason to believe that chemical affinity passes in its lowest degree into the attraction of aggregation.”[505]

[505] *Journal of the Chemical Society*, vol. viii. p. 51.

The atomic theory is well established, but its limits are not marked out. As Grove points out, we may by selecting sufficiently high multipliers express any combination or mixture of elements in terms of their equivalent weights.[506] Sir W. Thomson has suggested that the power which vegetable fibre, oatmeal, and other substances possess of attracting and condensing aqueous vapour is probably continuous, or, in fact, identical with capillary attraction, which is capable of interfering with the pressure of aqueous vapour and aiding its condensation.[507] There are many cases of so-called catalytic or surface action, such as the extraordinary power of animal charcoal for attracting organic matter, or of spongy platinum for condensing hydrogen, which can only be considered as exalted cases of a more general power of attraction. The number of substances which are decomposed by light in a striking manner is very limited; but many other substances, such as vegetable colours, are affected by long exposure; on the principle of continuity we might expect to find that all kinds of matter are more or less susceptible of change by the incidence of light rays.[508] It is the opinion of Grove that wherever an electric current passes there is a tendency to decomposition, a strain on the molecules, which when sufficiently intense leads to disruption. Even a metallic conducting wire may be regarded as tending to decomposition. Davy was probably correct in describing electricity as chemical affinity acting on masses, or rather, as Grove suggests, creating a disturbance through a chain of particles.[509] Laplace went so far as to suggest that all chemical phenomena may be results of the Newtonian law of attraction, applied to atoms of various mass and position; but the time is probably far distant when the progress of molecular philosophy and of mathematical methods will enable such a generalisation to be verified or refuted.

[506] *Correlation of Physical Forces*, 3rd edit. p. 184.

[507] *Philosophical Magazine*, 4th Series, vol. xlii. p. 451.

[508] Grove, *Correlation of Physical Forces*, 3rd edit. p. 118.

[509] Ibid. pp. 166, 199, &c.

*The Law of Continuity.*

Under the title of the Law of Continuity we may place many applications of the general principle of reasoning, that what is true of one case will be true of similar cases, and probably true of what are probably similar. Whenever we find that a law or similarity is rigorously fulfilled up to a certain point in time or space, we expect with a high degree of probability that it will continue to be fulfilled at least a little further. If we see part only of a circle, we naturally expect that the circular form will be continued in the part hidden from us. If a body has moved uniformly over a certain space, we expect that it will continue to move uniformly. The ground of such inferences is doubtless identical with that of other inductive inferences. In continuous motion every infinitely small space passed over constitutes a separate constituent fact, and had we perfect powers of observation the smallest finite motion would include an infinity of information, which, by the principles of the inverse method of probabilities, would enable us to infer with certainty to the next infinitely small portion of the path. But when we attempt to infer from one finite portion of a path to another finite portion, inference will be only more or less probable, according to the comparative lengths of the portions and the accuracy of observation; the longer our experience is, the more probable our inference will be; the greater the length of time or space over which the inference extends, the less probable.

This principle of continuity presents itself in nature in a great variety of forms and cases. It is familiarly expressed in the dictum *Natura non agit per saltum*. As Graham expressed the maxim, there are in nature no abrupt transitions, and the distinctions of class are never absolute.[510] There is always some notice--some forewarning of every phenomenon, and every change begins by insensible degrees, could we observe it with perfect accuracy. The cannon ball, indeed, is forced from the cannon in an inappreciable portion of time; the trigger is pulled, the fuze fired, the powder inflamed, the ball expelled, all simultaneously to our senses. But there is no doubt that time is occupied by every part of the process, and that the ball begins to move at first with infinite slowness. Captain Noble is able to measure by his chronoscope the progress of the shot in a 300-pounder gun, and finds that the whole motion within the barrel takes place in something less than one 200th part of a second. It is certain that no finite force can produce motion, except in a finite space of time. The amount of momentum communicated to a body is proportional to the accelerating force multiplied by the time during which it acts uniformly. Thus a slight force produces a great velocity only by long-continued action. In a powerful shock, like that of a railway collision, the stroke of a hammer on an anvil, or the discharge of a gun, the time is very short, and therefore the accelerating forces brought into play are exceedingly great, but never infinite. In the case of a large gun the powder in exploding is said to exert for a moment a force equivalent to at least 2,800,000 horses.

[510] *Philosophical Transactions*, 1861. *Chemical and Physical Researches*, p. 598.

Our belief in some of the fundamental laws of nature rests upon the principle of continuity. Galileo is held to be the first philosopher who consciously employed this principle in his arguments concerning the nature of motion, and it is certain that we can never by mere experience assure ourselves of the truth even of the first law of motion. *A material particle*, we are told, *when not acted on by extraneous forces will continue in the same state of rest or motion.* This may be true, but as we can find no body which is free from the action of extraneous causes, how are we to prove it? Only by observing that the less the amount of those forces the more nearly is the law found to be true. A ball rolled along rough ground is soon stopped; along a smooth pavement it continues longer in movement. A delicately suspended pendulum is almost free from friction against its supports, but it is gradually stopped by the resistance of the air; place it in the vacuous receiver of an air-pump and we find the motion much prolonged. A large planet like Jupiter experiences almost infinitely less friction, in comparison to its vast momentum, than we can produce experimentally, and we find in such a case that there is not the least evidence of the falsity of the law. Experience, then, informs us that we may approximate indefinitely to a uniform motion by sufficiently decreasing the disturbing forces. It is an act of inference which enables us to travel on beyond experience, and assert that, in the total absence of any extraneous force, motion would be absolutely uniform. The state of rest, again, is a limiting case in which motion is infinitely small or zero, to which we may attain, on the principle of continuity, by successively considering cases of slower and slower motion. There are many classes of phenomena, in which, by gradually passing from the apparent to the obscure, we can assure ourselves of the nature of phenomena which would otherwise be a matter of great doubt. Thus we can sufficiently prove in the manner of Galileo, that a musical sound consists of rapid uniform pulses, by causing strokes to be made at intervals which we gradually diminish until the separate strokes coalesce into a uniform hum or note. With great advantage we approach, as Tyndall says, the sonorous through the grossly mechanical. In listening to a great organ we cannot fail to perceive that the longest pipes, or their partial tones, produce a tremor and fluttering of the building. At the other extremity of the scale, there is no fixed limit to the acuteness of sounds which we can hear; some individuals can hear sounds too shrill for other ears, and as there is nothing in the nature of the atmosphere to prevent the existence of undulations far more rapid than any of which we are conscious, we may infer, by the principle of continuity, that such undulations probably exist.

There are many habitual actions which we perform we know not how. So rapidly are acts of minds accomplished that analysis seems impossible. We can only investigate them when in process of formation, observing that the best formed habit is slowly and continuously acquired, and it is in the early stages that we can perceive the rationale of the process.

Let it be observed that this principle of continuity must be held of much weight only in exact physical laws, those which doubtless repose ultimately upon the simple laws of motion. If we fearlessly apply the principle to all kinds of phenomena, we may often be right in our inferences, but also often wrong. Thus, before the development of spectrum analysis, astronomers had observed that the more they increased the powers of their telescopes the more nebulæ they could resolve into distinct stars. This result had been so often found true that they almost irresistibly assumed that all nebulæ would be ultimately resolved by telescopes of sufficient power; yet Huggins has in recent years proved by the spectroscope, that certain nebulæ are actually gaseous, and in a truly nebulous state.

The principle of continuity must have been continually employed in the inquiries of Galileo, Newton, and other experimental philosophers, but it appears to have been distinctly formulated for the first time by Leibnitz. He at least claims to have first spoken of “the law of continuity” in a letter to Bayle, printed in the *Nouvelles de la République des Lettres*, an extract from which is given in Erdmann’s edition of Leibnitz’s works, p. 104, under the title “Sur un Principe Général utile à l’explication des Lois de la Nature.”[511] It has indeed been asserted that the doctrine of the *latens processus* of Francis Bacon involves the principle of continuity,[512] but I think that this doctrine, like that of the *natures* of substances, is merely a vague statement of the principle of causation.

[511] *Life of Sir W. Hamilton*, p. 439.

[512] Powell’s *History of Natural Philosophy*, p. 201. *Novum Organum*, bk. ii. Aphorisms 5–7.

*Failure of the Law of Continuity.*

There are certain cautions which must be given as to the application of the principle of continuity. In the first place, where this principle really holds true, it may seem to fail owing to our imperfect means of observation. Though a physical law may not admit of perfectly abrupt change, there is no limit to the approach which it may make to abruptness. When we warm a piece of very cold ice, the absorption of heat, the temperature, and the dilatation of the ice vary according to apparently simple laws until we come to the zero of the Centigrade scale. Everything is then changed; an enormous absorption of heat takes place without any rise of temperature, and the volume of the ice decreases as it changes into water. Unless carefully investigated, this change appears to be perfectly abrupt; but accurate observation seems to show that there is a certain forewarning; the ice does not turn into water all at once, but through a small fraction of a degree the change is gradual. All the phenomena concerned, if measured very exactly, would be represented not by angular lines, but continuous curves, undergoing rapid flexures; and we may probably assert with safety that between whatever points of temperature we examine ice, there would be found some indication, though almost infinitesimally small, of the apparently abrupt change which was to occur at a higher temperature. It might also be pointed out that the important and apparently simple physical laws, such as those of Boyle and Mariotte, Dalton and Gay-Lussac, &c., are only approximately true, and the divergences from the simple laws are forewarnings of abrupt changes, which would otherwise break the law of continuity.

Secondly, it must be remembered that mathematical laws of some complexity will probably present singular cases or negative results, which may bear the appearance of discontinuity, as when the law of retraction suddenly yields us with perfect abruptness the phenomenon of total internal reflection. In the undulatory theory, however, there is no real change of law between refraction and reflection. Faraday in the earlier part of his career found so many substances possessing magnetic power, that he ventured on a great generalisation, and asserted that all bodies shared in the magnetic property of iron. His mistake, as he afterwards discovered, consisted in overlooking the fact that though magnetic in a certain sense, some substances have negative magnetism, and are repelled instead of being attracted by the magnet.

Thirdly, where we might expect to find a uniform mathematical law prevailing, the law may undergo abrupt change at singular points, and actual discontinuity may arise. We may sometimes be in danger of treating under one law phenomena which really belong to different laws. For instance, a spherical shell of uniform matter attracts an external particle of matter with a force varying inversely as the square of the distance from the centre of the sphere. But this law only holds true so long as the particle is external to the shell. Within the shell the law is wholly different, and the aggregate gravity of the sphere becomes zero, the force in every direction being neutralised by an exactly equal opposite force. If an infinitely small particle be in the superficies of a sphere, the law is again different, and the attractive power of the shell is half what it would be with regard to particles infinitely close to the surface of the shell. Thus in approaching the centre of a shell from a distance, the force of gravity shows double discontinuity in passing through the shell.[513]

[513] Thomson and Tait, *Treatise on Natural Philosophy*, vol. i. pp. 346–351.

It may admit of question, too, whether discontinuity is really unknown in nature. We perpetually do meet with events which are real breaks upon the previous law, though the discontinuity may be a sign that some independent cause has come into operation. If the ordinary course of the tides is interrupted by an enormous irregular wave, we attribute it to an earthquake, or some gigantic natural disturbance. If a meteoric stone falls upon a person and kills him, it is clearly a discontinuity in his life, of which he could have had no anticipation. A sudden sound may pass through the air neither preceded nor followed by any continuous effect. Although, then, we may regard the Law of Continuity as a principle of nature holding rigorously true in many of the relations of natural forces, it seems to be a matter of difficulty to assign the limits within which the law is verified. Much caution is required in its application.

*Negative Arguments on the Principle of Continuity.*

Upon the principle of continuity we may sometimes found arguments of great force which prove an hypothesis to be impossible, because it would involve a continual repetition of a process *ad infinitum*, or else a purely arbitrary breach at some point. Bonnet’s famous theory of reproduction represented every living creature as containing germs which were perfect representatives of the next generation, so that on the same principle they necessarily included germs of the next generation, and so on indefinitely. The theory was sufficiently refuted when once clearly stated, as in the following poem called the Universe,[514] by Henry Baker:--

“Each seed includes a plant: that plant, again, Has other seeds, which other plants contain: Those other plants have all their seeds, and those More plants again, successively inclose.

“Thus, ev’ry single berry that we find, Has, really, in itself whole forests of its kind, Empire and wealth one acorn may dispense, By fleets to sail a thousand ages hence.”

[514] *Philosophical Transactions* (1740), vol. xli. p. 454.

The general principle of inference, that what we know of one case must be true of similar cases, so far as they are similar, prevents our asserting anything which we cannot apply time after time under the same circumstances. On this principle Stevinus beautifully demonstrated that weights resting on two inclined planes and balancing each other must be proportional to the lengths of the planes between their apex and a horizontal plane. He imagined a uniform endless chain to be hung over the planes, and to hang below in a symmetrical festoon. If the chain were ever to move by gravity, there would be the same reason for its moving on for ever, and thus producing a perpetual motion. As this is absurd, the portions of the chain lying on the planes, and equal in length to the planes, must balance each other. On similar grounds we may disprove the existence of any *self-moving machine*; for if it could once alter its own state of motion or rest, in however small a degree, there is no reason why it should not do the like time after time *ad infinitum*. Newton’s proof of his third law of motion, in the case of gravity, is of this character. For he remarks that if two gravitating bodies do not exert exactly equal forces in opposite directions, the one exerting the strongest pull will carry both away, and the two bodies will move off into space together with velocity increasing *ad infinitum*. But though the argument might seem sufficiently convincing, Newton in his characteristic way made an experiment with a loadstone and iron floated upon the surface of water.[515] In recent years the very foundation of the principle of conservation of energy has been placed on the assumption that it is impossible by any combination of natural bodies to produce force continually from nothing.[516] The principle admits of application in various subtle forms.

[515] *Principia*, bk. i. Law iii. Corollary 6.

[516] Helmholtz, Taylor’s *Scientific Memoirs* (1853), vol. vi. p. 118.

Lucretius attempted to prove, by a most ingenious argument of this kind, that matter must be indestructible. For if a finite quantity, however small, were to fall out of existence in any finite time, an equal quantity might be supposed to lapse in every equal interval of time, so that in the infinity of past time the universe must have ceased to exist.[517] But the argument, however ingenious, seems to fail at several points. If past time be infinite, why may not matter have been created infinite also? It would be most reasonable, again, to suppose the matter destroyed in any time to be proportional to the matter then remaining, and not to the original quantity; under this hypothesis even a finite quantity of original matter could never wholly disappear from the universe. For like reasons we cannot hold that the doctrine of the conservation of energy is really proved, or can ever be proved to be absolutely true, however probable it may be regarded.

[517] *Lucretius*, bk. i. lines 232–264.

*Tendency to Hasty Generalisation.*

In spite of all the powers and advantages of generalisation, men require no incitement to generalise; they are too apt to draw hasty and ill-considered inferences. As Francis Bacon said, our intellects want not wings, but rather weights of lead to moderate their course.[518] The process is inevitable to the human mind; it begins with childhood and lasts through the second childhood. The child that has once been hurt fears the like result on all similar occasions, and can with difficulty be made to distinguish between case and case. It is caution and discrimination in the adoption of conclusions that we have chiefly to learn, and the whole experience of life is one continued lesson to this effect. Baden Powell has excellently described this strong natural propensity to hasty inference, and the fondness of the human mind for tracing resemblances real or fanciful. “Our first inductions,” he says,[519] “are always imperfect and inconclusive; we advance towards real evidence by successive approximations; and accordingly we find false generalisation the besetting error of most first attempts at scientific research. The faculty to generalise accurately and philosophically requires large caution and long training, and is not fully attained, especially in reference to more general views, even by some who may properly claim the title of very accurate scientific observers in a more limited field. It is an intellectual habit which acquires immense and accumulating force from the contemplation of wider analogies.”

[518] *Novum Organum*, bk. 1 Aphorism 104.

[519] *The Unity of Worlds and of Nature*, 2nd edit. p. 116.

Hasty and superficial generalisations have always been the bane of science, and there would be no difficulty in finding endless illustrations. Between things which are the same in number there is a certain resemblance, namely in number; but in the infancy of science men could not be persuaded that there was not a deeper resemblance implied in that of number. Pythagoras was not the inventor of a mystical science of number. In the ancient Oriental religions the seven metals were connected with the seven planets, and in the seven days of the week we still have, and probably always shall have, a relic of the septiform system ascribed by Dio Cassius to the ancient Egyptians. The disciples of Pythagoras carried the doctrine of the number seven into great detail. Seven days are mentioned in Genesis; infants acquire their teeth at the end of seven months; they change them at the end of seven years; seven feet was the limit of man’s height; every seventh year was a climacteric or critical year, at which a change of disposition took place. Then again there were the seven sages of Greece, the seven wonders of the world, the seven rites of the Grecian games, the seven gates of Thebes, and the seven generals destined to conquer that city.

In natural science there were not only the seven planets, and the seven metals, but also the seven primitive colours, and the seven tones of music. So deep a hold did this doctrine take that we still have its results in many customs, not only in the seven days of the week, but the seven years’ apprenticeship, puberty at fourteen years, the second climacteric, and legal majority at twenty-one years, the third climacteric. The idea was reproduced in the seven sacraments of the Roman Catholic Church, and the seven year periods of Comte’s grotesque system of domestic worship. Even in scientific matters the loftiest intellects have occasionally yielded, as when Newton was misled by the analogy between the seven tones of music and the seven colours of his spectrum. Other numerical analogies, though rejected by Galileo, held Kepler in thraldom; no small part of Kepler’s labours during seventeen years was spent upon numerical and geometrical analogies of the most baseless character; and he gravely held that there could not be more than six planets, because there were not more than five regular solids. Even the genius of Huyghens did not prevent him from inferring that but one satellite could belong to Saturn, because, with those of Jupiter and the Earth, it completed the perfect number of six. A whole series of other superstitions and fallacies attach to the numbers six and nine.

It is by false generalisation, again, that the laws of nature have been supposed to possess that perfection which we attribute to simple forms and relations. The heavenly bodies, it was held, must move in circles, for the circle was the perfect figure. Newton seemed to adopt the questionable axiom that nature always proceeds in the simplest way; in stating his first rule of philosophising, he adds:[520] “To this purpose the philosophers say, that nature does nothing in vain, when less will serve; for nature is pleased with simplicity, and affects not the pomp of superfluous causes.” Keill lays down[521] as an axiom that “The causes of natural things are such, as are the most simple, and are sufficient to explain the phenomena: for nature always proceeds in the simplest and most expeditious method; because by this manner of operating the Divine Wisdom displays itself the more.” If this axiom had any clear grounds of truth, it would not apply to proximate laws; for even when the ultimate law is simple the results may be infinitely diverse, as in the various elliptic, hyperbolic, parabolic, or circular orbits of the heavenly bodies. Simplicity is naturally agreeable to a mind of limited powers, but to an infinite mind all things are simple.

[520] *Principia*, bk. iii, *ad initium*.

[521] Keill, *Introduction to Natural Philosophy*, p. 89.

Every great advance in science consists in a great generalisation, pointing out deep and subtle resemblances. The Copernican system was a generalisation, in that it classed the earth among the planets; it was, as Bishop Wilkins expressed it, “the discovery of a new planet,” but it was opposed by a more shallow generalisation. Those who argued from the condition of things upon the earth’s surface, thought that every object must be attached to and rest upon something else. Shall the earth, they said, alone be free? Accustomed to certain special results of gravity they could not conceive its action under widely different circumstances.[522] No hasty thinker could seize the deep analogy pointed out by Horrocks between a pendulum and a planet, true in substance though mistaken in some details. All the advances of modern science rise from the conception of Galileo, that in the heavenly bodies, however apparently different their condition, we shall ultimately recognise the same fundamental principles of mechanical science which are true on earth.

[522] Jeremiæ Horroccii *Opera Posthuma* (1673), pp. 26, 27.

Generalisation is the great prerogative of the intellect, but it is a power only to be exercised safely with much caution and after long training. Every mind must generalise, but there are the widest differences in the depth of the resemblances discovered and the care with which the discovery is verified. There seems to be an innate power of insight which a few men have possessed pre-eminently, and which enabled them, with no exemption indeed from labour or temporary error, to discover the one in the many. Minds of excessive acuteness may exist, which have yet only the powers of minute discrimination, and of storing up, in the treasure-house of memory, vast accumulations of words and incidents. But the power of discovery belongs to a more restricted class of minds. Laplace said that, of all inventors who had contributed the most to the advancement of human knowledge, Newton and Lagrange appeared to possess in the highest degree the happy tact of distinguishing general principles among a multitude of objects enveloping them, and this tact he conceived to be the true characteristic of scientific genius.[523]

[523] Young’s *Works*, vol. ii. p. 564.