CHAPTER XXVIII.
ANALOGY.
As we have seen in the previous chapter, generalisation passes insensibly into reasoning by analogy, and the difference is one of degree. We are said to generalise when we view many objects as agreeing in a few properties, so that the resemblance is extensive rather than deep. When we have only a few objects of thought, but are able to discover many points of resemblance, we argue by analogy that the correspondence will be even deeper than appears. It may not be true that the words are always used in such distinct senses, and there is great vagueness in the employment of these and many logical terms; but if any clear discrimination can be drawn between generalisation and analogy, it is as indicated above.
It has been said, indeed, that analogy denotes not a resemblance between things, but between the relations of things. A pilot is a very different man from a prime minister, but he bears the same relation to a ship that the minister does to the state, so that we may analogically describe the prime minister as the pilot of the state. A man differs still more from a horse, nevertheless four men bear to three men the same relation as four horses bear to three horses. There is a real analogy between the tones of the Monochord, the Sages of Greece, and the Gates of Thebes, but it does not extend beyond the fact that they were all seven in number. Between the most discrete notions, as, for instance, those of time and space, analogy may exist, arising from the fact that the mathematical conditions of the lapse of time and of motion along a line are similar. There is no identity of nature between a word and the thing it signifies; the substance *iron* is a heavy solid, the word *iron* is either a momentary disturbance of the air, or a film of black pigment on white paper; but there is analogy between words and their significates. The substance iron is to the substance iron-carbonate, as the name iron is to the name iron-carbonate, when these names are used according to their scientific definitions. The whole structure of language and the whole utility of signs, marks, symbols, pictures, and representations of various kinds, rest upon analogy. I may hope perhaps to enter more fully upon this important subject at some future time, and to attempt to show how the invention of signs enables us to express, guide, and register our thoughts. It will be sufficient to observe here that the use of words constantly involves analogies of a subtle kind; we should often be at a loss how to describe a notion, were we not at liberty to employ in a metaphorical sense the name of anything sufficiently resembling it. There would be no expression for the sweetness of a melody, or the brilliancy of an harangue, unless it were furnished by the taste of honey and the brightness of a torch.
A cursory examination of the way in which we popularly use the word analogy, shows that it includes all degrees of resemblance or similarity. The analogy may consist only in similarity of number or ratio, or in like relations of time and space. It may also consist in simple resemblance between physical properties. We should not be using the word inconsistently with custom, if we said that there was an analogy between iron, nickel, and cobalt, manifested in the strength of their magnetic powers. There is a still more perfect analogy between iodine and chlorine; not that every property of iodine is identical with the corresponding property of chlorine; for then they would be one and the same kind of substance, and not two substances; but every property of iodine resembles in all but degree some property of chlorine. For almost every substance in which iodine forms a component, a corresponding substance may be discovered containing chlorine, so that we may confidently infer from the compounds of the one to the compounds of the other substance. Potassium iodide crystallises in cubes; therefore it is to be expected that potassium chloride will also crystallise in cubes. The science of chemistry as now developed rests almost entirely upon a careful and extensive comparison of the properties of substances, bringing deep-lying analogies to light. When any new substance is encountered, the chemist is guided in his treatment of it by the analogies which it seems to present with previously known substances.
In this chapter I cannot hope to illustrate the all-pervading influence of analogy in human thought and science. All science, it has been said, at the outset, arises from the discovery of identity, and analogy is but one name by which we denote the deeper-lying cases of resemblance. I shall only try to point out at present how analogy between apparently diverse classes of phenomena often serves as a guide in discovery. We thus commonly gain the first insight into the nature of an apparently unique object, and thus, in the progress of a science, we often discover that we are treating over again, in a new form, phenomena which were well known to us in another form.
*Analogy as a Guide in Discovery.*
There can be no doubt that discovery is most frequently accomplished by following up hints received from analogy, as Jeremy Bentham remarked.[524] Whenever a phenomenon is perceived, the first impulse of the mind is to connect it with the most nearly similar phenomenon. If we could ever meet a thing wholly *sui generis*, presenting no analogy to anything else, we should be incapable of investigating its nature, except by purely haphazard trial. The probability of success by such a process is so slight, that it is preferable to follow up the faintest clue. As I have pointed out already (p. 418), the possible experiments are almost infinite in number, and very numerous also are the hypotheses upon which we may proceed. Now it is self-evident that, however slightly superior the probability of success by one course of procedure may be over another, the most probable one should always be adopted first.
[524] *Essay on Logic*, *Works*, vol. viii. p. 276.
The chemist having discovered what he believes to be a new element, will have before him an infinite variety of modes of treating and investigating it. If in any of its qualities the substance displays a resemblance to an alkaline metal, for instance, he will naturally proceed to try whether it possesses other properties of the alkaline metals. Even the simplest phenomenon presents so many points for notice that we have a choice from among many hypotheses.
It would be difficult to find a more instructive instance of the way in which the mind is guided by analogy than in the description by Sir John Herschel of the course of thought by which he was led to anticipate in theory one of Faraday’s greatest discoveries. Herschel noticed that a screw-like form, technically called helicoidal dissymmetry, was observed in three cases, namely, in electrical helices, plagihedral quartz crystals, and the rotation of the plane of polarisation of light. As he said,[525] “I reasoned thus: Here are three phenomena agreeing in a *very strange peculiarity*. Probably, this peculiarity is a connecting link, physically speaking, among them. Now, in the case of the crystals and the light, this probability has been turned into certainty by my own experiments. Therefore, induction led me to conclude that a similar connection exists, and must turn up, somehow or other, between the electric current and polarised light, and that the plane of polarisation would be deflected by magneto-electricity.” By this course of analogical thought Herschel had actually been led to anticipate Faraday’s great discovery of the influence of magnetic strain upon polarised light. He had tried in 1822–25 to discover the influence of electricity on light, by sending a ray of polarised light through a helix, or near a long wire conveying an electric current. Such a course of inquiry, followed up with the persistency of Faraday, and with his experimental resources, would doubtless have effected the discovery. Herschel also suggests that the plagihedral form of quartz crystals must be due to a screw-like strain during crystallisation; but the notion remains unverified by experiment.
[525] *Life of Faraday*, by Bence Jones, vol. ii. p. 206.
*Analogy in the Mathematical Sciences.*
Whoever wishes to acquire a deep acquaintance with Nature must observe that there are analogies which connect whole branches of science in a parallel manner, and enable us to infer of one class of phenomena what we know of another. It has thus happened on several occasions that the discovery of an unsuspected analogy between two branches of knowledge has been the starting-point for a rapid course of discovery. The truths readily observed in the one may be of a different character from those which present themselves in the other. The analogy, once pointed out, leads us to discover regions of one science yet undeveloped, to which the key is furnished by the corresponding truths in the other science. An interchange of aid most wonderful in its results may thus take place, and at the same time the mind rises to a higher generalisation, and a more comprehensive view of nature.
No two sciences might seem at first sight more different in their subject matter than geometry and algebra. The first deals with circles, squares, parallelograms, and other forms in space; the latter with mere symbols of number. Prior to the time of Descartes, the sciences were developed slowly and painfully in almost entire independence of each other. The Greek philosophers indeed could not avoid noticing occasional analogies, as when Plato in the Thæetetus describes a square number as *equally equal*, and a number produced by multiplying two unequal factors as *oblong*. Euclid, in the 7th and 8th books of his Elements, continually uses expressions displaying a consciousness of the same analogies, as when he calls a number of two factors a *plane number*, ἐπίπεδος ἀριθμός, and distinguishes a square number of which the two factors are equal as an equal-sided and plane number, ἰσόπλευρος καὶ ἐπίπεδος ἀριθμός. He also calls the root of a cubic number its side, πλευρά. In the Diophantine algebra many problems of a geometrical character were solved by algebraic or numerical processes; but there was no general system, so that the solutions were of an isolated character. In general the ancients were far more advanced in geometric than symbolic methods; thus Euclid in his 4th book gives the means of dividing a circle by purely geometric means into 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30 parts, but he was totally unacquainted with the theory of the roots of unity exactly corresponding to this division of the circle.
During the middle ages, on the contrary, algebra advanced beyond geometry, and modes of solving equations were gradually discovered by those who had no notion that at every step they were implicitly solving geometric problems. It is true that Regiomontanus, Tartaglia, Bombelli, and possibly other early algebraists, solved isolated geometrical problems by the aid of algebra, but particular numbers were always used, and no consciousness of a general method was displayed. Vieta in some degree anticipated the final discovery, and occasionally represented the roots of an equation geometrically, but it was reserved for Descartes to show, in the most general manner, that every equation may be represented by a curve or figure in space, and that every bend, point, cusp, or other peculiarity in the curve indicates some peculiarity in the equation. It is impossible to describe in any adequate manner the importance of this discovery. The advantage was two-fold: algebra aided geometry, and geometry gave reciprocal aid to algebra. Curves such as the well-known sections of the cone were found to correspond to quadratic equations; and it was impossible to manipulate the equations without discovering properties of those all-important curves. The way was thus opened for the algebraic treatment of motions and forces, without which Newton’s *Principia* could never have been worked out. Newton indeed was possessed by a strong infatuation in favour of the ancient geometrical methods; but it is well known that he employed symbolic methods to discover his theorems, and he now and then, by some accidental use of algebraic expression, confessed its greater power and generality.
Geometry, on the other hand, gave great assistance to algebra, by affording concrete representations of relations which would otherwise be too abstract for easy comprehension. A curve of no great complexity may give the whole history of the variations of value of a troublesome mathematical expression. As soon as we know, too, that every regular geometrical curve represents some algebraic equation, we are presented by observation of mechanical movements with abundant suggestions towards the discovery of mathematical problems. Every particle of a carriage-wheel when moving on a level road is constantly describing a cycloidal curve, the curious properties of which exercised the ingenuity of all the most skilful mathematicians of the seventeenth century, and led to important advancements in algebraic power. It may be held that the discovery of the Differential Calculus was mainly due to geometrical analogy, because mathematicians, in attempting to treat algebraically the tangent of a curve, were obliged to entertain the notion of infinitely small quantities.[526] There can be no doubt that Newton’s fluxional, that is, geometrical mode of stating the differential calculus, however much it subsequently retarded its progress in England, facilitated its apprehension at first, and I should think it almost certain that Newton discovered the principles of the calculus geometrically.
[526] Lacroix, *Traité Élémentaire de Calcul Différentiel et de Calcul Intégral*, 5^{me} édit. p. 699.
We may accordingly look upon this discovery of analogy, this happy alliance, as Bossut calls it,[527] between geometry and algebra, as the chief source of discoveries which have been made for three centuries past in mathematical methods. This is certainly the opinion of Lagrange, who says, “So long as algebra and geometry have been separate, their progress was slow, and their employment limited; but since these two sciences have been united, they have lent each other mutual strength, and have marched together with a rapid step towards perfection.”
[527] *Histoire des Mathématiques*, vol. i. p. 298.
The advancement of mechanical science has also been greatly aided by analogy. An abstract and intangible existence like force demands much power of conception, but it has a perfect concrete representative in a line, the end of which may denote the point of application, and the direction the line of action of the force, while the length can be made arbitrarily to denote the amount of the force. Nor does the analogy end here; for the moment of the force about any point, or its product into the perpendicular distance of its line of action from the point, is found to be represented by an area, namely twice the area of the triangle contained between the point and the ends of the line representing the force. Of late years a great generalisation has been effected; the Double Algebra of De Morgan is true not only of space relations, but of forces, so that the triangle of forces is reduced to a case of pure geometrical addition. Nay, the triangle of lines, the triangle of velocities, the triangle of forces, the triangle of couples, and perhaps other cognate theorems, are reduced by analogy to one simple theorem, which amounts to this, that there are two ways of getting from one angular point of a triangle to another, which ways, though different in length, are identical in their final results.[528] In the system of quaternions of the late Sir W. R. Hamilton, these analogies are embodied and carried out in the most general manner, so that whatever problem involves the threefold dimensions of space, or relations analogous to those of space, is treated by a symbolic method of the most comprehensive simplicity.
[528] See Goodwin, *Cambridge Philosophical Transactions* (1845), vol. viii. p. 269. O’Brien, “On Symbolical Statics,” *Philosophical Magazine*, 4th Series, vol. i. pp. 491, &c. See also Professor Clerk Maxwell’s delightful *Manual of Elementary Science*, called *Matter and Motion*, published by the Society for Promoting Christian Knowledge. In this admirable little work some of the most advanced results of mechanical and physical science are explained according to the method of quaternions, but with hardly any use of algebraic symbols.
It ought to be added that to the discovery of analogy between the forms of mathematical and logical expressions, we owe the greatest advance in logical science. Boole based his extension of logical processes upon the notion that logic is an algebra of two quantities 0 and 1. His profound genius for symbolic investigation led him to perceive by analogy that there must exist a general system of logical deduction, of which the old logicians had seized only a few fragments. Mistaken as he was in placing algebra as a higher science than logic, no one can deny that the development of the more complex and dependent science had advanced far beyond that of the simpler science, and that Boole, in drawing attention to the connection, made one of the most important discoveries in the history of science. As Descartes had wedded algebra and geometry, so did Boole accomplish the marriage of logic and algebra.
*Analogy in the Theory of Undulations.*
There is no class of phenomena which more thoroughly illustrates alike the power and weakness of analogy than the waves which agitate every kind of medium. All waves, whatsoever be the matter through which they pass, obey the principles of rhythmical or harmonic motion, and the subject therefore presents a fine field for mathematical generalisation. Each kind of medium may allow of waves peculiar in their conditions, so that it is a beautiful exercise in analogical reasoning to decide how, in making inferences from one kind of medium to another, we must make allowance for difference of circumstances. The waves of the ocean are large and visible, and there are the yet greater tidal waves which extend around the globe. From such palpable cases of rhythmical movement we pass to waves of sound, varying in length from about 32 feet to a small fraction of an inch. We have but to imagine, if we can, the fortieth octave of the middle C of a piano, and we reach the undulations of yellow light, the ultra-violet being about the forty-first octave. Thus we pass from the palpable and evident to that which is obscure, if not incomprehensible. Yet the same phenomena of reflection, interference, and refraction, which we find in some kinds of waves, may be expected to occur, *mutatis mutandis*, in other kinds.
From the great to the small, from the evident to the obscure, is not only the natural order of inference, but it is the historical order of discovery. The physical science of the Greek philosophers must have remained incomplete, and their theories groundless, because they did not understand the nature of undulations. Their systems were based upon the notion of movement of translation from place to place. Modern science tends to the opposite notion that all motion is alternating or rhythmical, energy flowing onwards but matter remaining comparatively fixed in position. Diogenes Laertius indeed correctly compared the propagation of sound with the spreading of waves on the surface of water when disturbed by a stone, and Vitruvius displayed a more complete comprehension of the same analogy. It remained for Newton to create the theory of undulatory motion in showing by mathematical deductive reasoning that the particles of an elastic fluid by vibrating backwards and forwards, might carry a pulse or wave moving from the source of disturbance, while the disturbed particles return to their place of rest. He was even able to make a first approximation by theoretical calculation to the velocity of sound-waves in the atmosphere. His theory of sound formed a hardly less important epoch in science than his far more celebrated theory of gravitation. It opened the way to all the subsequent applications of mechanical principles to the insensible motion of molecules. He seems to have been, too, upon the brink of another application of the same principles which would have advanced science by a century of progress, and made him the undisputed founder of all the theories of matter. He expressed opinions at various times that light might be due to undulatory movements of a medium occupying space, and in one intensely interesting sentence remarks[529] that colours are probably vibrations of different lengths, “much after the manner that, in the sense of hearing, nature makes use of aërial vibrations of several bignesses to generate sounds of divers tones, for the analogy of nature is to be observed.” He correctly foresaw that red and yellow light would consist of the longer undulations, and blue and violet of the shorter, while white light would be composed of an indiscriminate mixture of waves of various lengths. Newton almost overcame the strongest apparent difficulty of the undulatory theory of light, namely, the propagation of light in straight lines. For he observed that though waves of sound bend round an obstacle to some extent, they do not do so in the same degree as water-waves.[530] He had but to extend the analogy proportionally to light-waves, and not only would the difficulty have vanished, but the true theory of diffraction would have been open to him. Unfortunately he had a preconceived theory that rays of light are bent from and not towards the shadow of a body, a theory which for once he did not sufficiently compare with observation to detect its falsity. I am not aware, too, that Newton has, in any of his works, displayed an understanding of the phenomena of interference without which his notion of waves must have been imperfect.
[529] Birch, *History of the Royal Society*, vol. iii. p. 262, quoted by Young, *Works*, vol. i. p. 246.
[530] *Opticks*, Query 28, 3rd edit. p. 337.
While the general principles of undulatory motion will be the same in whatever medium the motion takes place, the circumstances may be excessively different. Between light travelling 186,000 miles per second and sound travelling in air only about 1,100 feet in the same time, or almost 900,000 times as slowly, we cannot expect a close outward resemblance. There are great differences, too, in the character of the vibrations. Gases scarcely admit of transverse vibration, so that sound travelling in air is a longitudinal wave, the particles of air moving backwards and forwards in the same line in which the wave moves onwards. Light, on the other hand, appears to consist entirely in the movement of points of force transversely to the direction of propagation of the ray. The light-wave is partially analogous to the bending of a rod or of a stretched cord agitated at one end. Now this bending motion may take place in any one of an infinite number of planes, and waves of which the planes are perpendicular to each other cannot interfere any more than two perpendicular forces can interfere. The complicated phenomena of polarised light arise out of this transverse character of the luminous wave, and we must not expect to meet analogous phenomena in atmospheric sound-waves. It is conceivable that in solids we might produce transverse sound undulations, in which phenomena of polarisation might be reproduced. But it would appear that even between transverse sound and light-waves the analogy holds true rather of the principles of harmonic motion than the circumstances of the vibrating medium; from experiment and theory it is inferred that the plane of polarisation in plane polarised light is perpendicular to instead of being coincident with the direction of vibration, as it would be in the case of transverse sound undulations. If so the laws of elastic forces are essentially different in application to the luminiferous ether and to ordinary solid bodies.[531]
[531] Rankine, *Philosophical Transactions* (1856), vol. cxlvi. p. 282.
*Analogy in Astronomy.*
We shall be much assisted in gaining a true appreciation of the value of analogy in its feebler degrees, by considering how much it has contributed to the progress of astronomical science. Our point of observation is so fixed with regard to the universe, and our means of examining distant bodies are so restricted, that we are necessarily guided by limited and apparently feeble resemblances. In many cases the result has been confirmed by subsequent direct evidence of the most forcible character.
While the scientific world was divided in opinion between the Copernican and Ptolemaic systems, it was analogy which furnished the most satisfactory argument. Galileo discovered, by the use of his new telescope, the four small satellites which circulate round Jupiter, and make a miniature planetary world. These four Medicean Stars, as they were called, were plainly seen to revolve round Jupiter in various periods, but approximately in one plane, and astronomers irresistibly inferred that what might happen on the smaller scale might also be found true of the greater planetary system. This discovery gave “the holding turn,” as Herschel expressed it, to the opinions of mankind. Even Francis Bacon, who, little to the credit of his scientific sagacity, had previously opposed the Copernican views, now became convinced, saying “We affirm the solisequium of Venus and Mercury; since it has been found by Galileo that Jupiter also has attendants.” Nor did Huyghens think it superfluous to adopt the analogy as a valid argument.[532] Even in an advanced stage of physical astronomy, the Jovian system has not lost its analogical interest; for the mutual perturbations of the four satellites pass through all their phases within a few centuries, and thus enable us to verify in a miniature case the principles of stability, which Laplace established for the great planetary system. Oscillations or disturbances which in the motions of the planets appear to be secular, because their periods extend over millions of years, can be watched, in the case of Jupiter’s satellites, through complete revolutions within the historical period of astronomy.[533]
[532] *Cosmotheoros* (1699), p. 16.
[533] Laplace, *System of the World*, vol. ii. p. 316.
In obtaining a knowledge of the stellar universe we must sometimes depend upon precarious analogies. We still hold upon this ground the opinion, entertained by Bruno as long ago as 1591, that the stars may be suns attended by planets like our earth. This is the most probable first assumption, and it is supported by spectrum observations, which show the similarity of light derived from many stars with that of the sun. But at the same time we learn by the prism that there are nebulæ and stars in conditions widely different from anything known in our system. In the course of time the analogy may perhaps be restored to comparative completeness by the discovery of suns in various stages of nebulous condensation. The history of the evolution of our own world may be traced back in bodies less developed, or traced forwards in systems more advanced towards the dissipation of energy, and the extinction of life. As in a great workshop, we may perhaps see the material work of Creation as it has progressed through thousands of millions of years.
In speculations concerning the physical condition of the planets and their satellites, we depend upon analogies of a weak character. We may be said to know that the moon has mountains and valleys, plains and ridges, volcanoes and streams of lava, and, in spite of the absence of air and water, the rocky surface of the moon presents so many familiar appearances that we do not hesitate to compare them with the features of our globe. We infer with high probability that Mars has polar snow and an atmosphere absorbing blue rays like our own; Jupiter undoubtedly possesses a cloudy atmosphere, possibly not unlike a magnified copy of that surrounding the earth, but our tendency to adopt analogies receives a salutary correction in the recently discovered fact that the atmosphere of Uranus contains hydrogen.
Philosophers have not stopped at these comparatively safe inferences, but have speculated on the existence of living creatures in other planets. Huyghens remarked that as we infer by analogy from the dissected body of a dog to that of a pig and ox or other animal of the same general form, and as we expect to find the same viscera, the heart, stomach, lungs, intestines, &c., in corresponding positions, so when we notice the similarity of the planets in many respects, we must expect to find them alike in other respects.[534] He even enters into an inquiry whether the inhabitants of other planets would possess reason and knowledge of the same sort as ours, concluding in the affirmative. Although the power of intellect might be different, he considers that they would have the same geometry if they had any at all, and that what is true with us would be true with them.[535] As regards the sun, he wisely observes that every conjecture fails. Laplace entertained a strong belief in the existence of inhabitants on other planets. The benign influence of the sun gives birth to animals and plants upon the surface of the earth, and analogy induces us to believe that his rays would tend to have a similar effect elsewhere. It is not probable that matter which is here so fruitful of life would be sterile upon so great a globe as Jupiter, which, like the earth, has its days and nights and years, and changes which indicate active forces. Man indeed is formed for the temperature and atmosphere in which he lives, and, so far as appears, could not live upon the other planets. But there might be an infinity of organisations relative to the diverse constitutions of the bodies of the universe. The most active imagination cannot form any idea of such various creatures, but their existence is not unlikely.[536]
[534] *Cosmotheoros* (1699), p. 17.
[535] Ibid. p. 36.
[536] *System of the World*, vol. ii. p. 326. *Essai Philosophique*, p. 87.
We now know that many metals and other elements never found in organic structures are yet capable of forming compounds with substances of vegetable or animal origin. It is therefore just possible that at different temperatures creatures formed of different yet analogous compounds might exist, but it would seem indispensable that carbon should form the basis of organic structures. We have no analogies to lead us to suppose that in the absence of that complex element life can exist. Could we find globes surrounded by atmospheres resembling our own in temperature and composition, we should be almost forced to believe them inhabited, but the probability of any analogical argument decreases rapidly as the condition of a globe diverges from that of our own. The Cardinal Nicholas de Cusa held long ago that the moon was inhabited, but the absence of any appreciable atmosphere renders the existence of inhabitants highly improbable. Speculations resting upon weak analogies hardly belong to the scope of true science, and can only be tolerated as an antidote to the far worse dogmas which assert that the thousand million of persons on earth, or rather a small fraction of them, are the sole objects of care of the Power which designed this limitless Universe.
*Failures of Analogy.*
So constant is the aid which we derive from the use of analogy in all attempts at discovery or explanation, that it is most important to observe in what cases it may lead us into difficulties. That which we expect by analogy to exist
(1) May be found to exist;
(2) May seem not to exist, but nevertheless may really exist;
(3) May actually be non-existent.
In the second case the failure is only apparent, and arises from our obtuseness of perception, the smallness of the phenomenon to be noticed, or the disguised character in which it appears. I have already pointed out that the analogy of sound and light seems to fail because light does not apparently bend round a corner, the fact being that it does so bend in the phenomena of diffraction, which present the effect, however, in such an unexpected and minute form, that even Newton was misled, and turned from the correct hypothesis of undulations which he had partially entertained.
In the third class of cases analogy fails us altogether, and we expect that to exist which really does not exist. Thus we fail to discover the phenomena of polarisation in sound travelling through the atmosphere, since air is not capable of any appreciable transverse undulations. These failures of analogy are of peculiar interest, because they make the mind aware of its superior powers. There have been many philosophers who said that we can conceive nothing in the intellect which we have not previously received through the senses. This is true in the sense that we cannot *image* them to the mind in the concrete form of a shape or a colour; but we can speak of them and reason concerning them; in short, we often know them in everything but a sensuous manner. Accurate investigation shows that all material substances retard the motion of bodies through them by subtracting energy by impact. By the law of continuity we can frame the notion of a vacuous space in which there is no resistance whatever, nor need we stop there; for we have only to proceed by analogy to the case where a medium should accelerate the motion of bodies passing through it, somewhat in the mode which Aristotelians attributed falsely to the air. Thus we can frame the notion of *negative density*, and Newton could reason exactly concerning it, although no such thing exists.[537]
[537] *Principia*, bk. ii. Section ii. Prop. x.
In every direction of thought we may meet ultimately with similar failures of analogy. A moving point generates a line, a moving line generates a surface, a moving surface generates a solid, but what does a moving solid generate? When we compare a polyhedron, or many-sided solid, with a polygon, or plane figure of many sides, the volume of the first is analogous to the area of the second; the face of the solid answers to the side of the polygon; the edge of the solid to the point of the figure; but the corner, or junction of edges in the polyhedron, is left wholly unrepresented in the plane of the polygon. Even if we attempted to draw the analogies in some other manner, we should still find a geometrical notion embodied in the solid which has no representative in the figure of two dimensions.[538]
[538] De Morgan, *Cambridge Philosophical Transactions*, vol. xi. Part ii. p. 246.
Faraday was able to frame some notion of matter in a fourth condition, which should be to gas what gas is to liquid.[539] Such substance, he thought, would not fall far short of *radiant matter*, by which apparently he meant the supposed caloric or matter assumed to constitute heat, according to the corpuscular theory. Even if we could frame the notion, matter in such a state cannot be known to exist, and recent discoveries concerning the continuity of the solid, liquid, and gaseous states remove the basis of the speculation.
[539] *Life of Faraday*, vol. i. p. 216.
From these and many other instances which might be adduced, we learn that analogical reasoning leads us to the conception of many things which, so far as we can ascertain, do not exist. In this way great perplexities have arisen in the use of language and mathematical symbols. All language depends upon analogy; for we join and arrange words so that they may represent the corresponding junctions or arrangements of things and their equalities. But in the use of language we are obviously capable of forming many combinations of words to which no corresponding meaning apparently exists. The same difficulty arises in the use of mathematical signs, and mathematicians have needlessly puzzled themselves about the square root of a negative quantity, which is, in many applications of algebraic calculation, simply a sign without any analogous meaning, there being a failure of analogy.