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: 1 "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 2 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. Those 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. 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. No hasty thinker could seize the deep analogy pointed out by Horrocks between a pen 1 Principia, bk. iii, ad initium. 2 Keill, Introduction to Natural Philosophy, p. 89. SS dulum 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. 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.1 1 Young's Works, vol. ii, p. 564 CHAPTER XXVIIL 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. 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 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 allpervading 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.1 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. 1 Essay on Logic, Works, vol. viii. p. 276. |