the Inverse Method of Probabilities (vol. i. pp. 27612), that whenever in the future we meet an object posssing either one of the properties of gravity and inertia, will be found on examination to possess the other of hese properties. This is a clear instance of the employment of generalization. In analogy, on the other hand, we reason from likeness n 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 would be found to consist of ice and snow, if we could examine them. 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 single 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 also 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'b. The nature b Essai Philosophique sur les Probabilitiés,' p. 86. of analogical inference is also very correctly described in the 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's, many (qualities) in one, therefore also the remainder in the same. It is evident that there may be intermediate cases in which, from the resemblance 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 versa. Two Meanings of Generalization. The term generalization, as commonly used, includes two processes which are of different character, but are often closely associated together. In the first place, we generalize whenever we recognise even in two facts or objects a certain 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 with a regular octahedral crystal, there is little apparent similarity; but, so soon as we perceive that either can be produced by the symmetrical modification of the other, we discover a groundwork of similarity in the constitution of 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 c Kant's 'Logik,' § 84, Königsberg, 1800, p. 207. Schönbein. There was a time when the rainbow was an entirely 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 other transparent stones. Drops of water scattered by the oar in the sun, the spray from a water-wheel, the dew-drops 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 generalize, 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 and phenomena, our comprehension of those already observed is vastly strengthened and deepened by thus learning to view them as particular cases of one more general property. A second process, to which the name of generalization is equally given, consists in passing from a given 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 generalization 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 or cooling, the more substances we can vapourize and freeze, we pass confidently d Whewell's Philosophy of the Inductive Sciences,' 2nd edit. vol. ii. p. 171, quoting the 'Opus Majus,' p. 473. in advance of fact, and assume that all substances are capable of these three forms. Such a generalization was accepted by men of the high intellect of Lavoisiere 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 at once considered sufficient indication that all comets must obey the same power. Few persons doubted that the same great law extended over the whole heavens; certainly the fact that a few stars out of many millions make manifest the action of gravity, is now held to be sufficient evidence to establish the general extension of the laws of Newton over the sphere of the visible universe. Value of Generalization. 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 into particular cases again? 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. e 6 Chemistry,' translated by Kerr, 3rd edit. pp. 63, 77. |