able that they often attached erroneous ideas to this agency. Yet Aristotle saw clearly enough that Chance was only a name for our ignorance of Cause; and could he or any other potent thinker of ancient times reappear, and listen to some discussions in our Academies, it is probable that he would be struck with the erroneous ideas now prevalent respecting Law. He would perhaps see that the conception, Law, was as much a realized abstraction as Chance; and might urge that Chance has the same claim to the position of a real agent as Law. Chance is a term by which we express the irregularities in phenomena, disregarding their uniformities; Law is a term by which we classify changes and express the uniformities in phenomena, disregarding their irregularities.* The phenomena themselves are uniform, in the sense of each being always what it is; they are irregular, in the sense of being conjoined now in one way and now in another.

91. When Philosophy first began to meditate on the various phenomena which incessantly presented themselves, it obeyed the identifying instinct which groups together resemblances, and gradually ranged these into separate classes. Objects were observed, and classified, according to their resemblances, in genera and species; changes were also observed, and classified in laws of Nature. A general conception of Order emerged in this separation of the like from the unlike. This conception rapidly became extended, owing to that tendency of the

* "Tous les événemens, ceux mêmes qui par leur petitesse semblent ne pas tenir aux grandes lois de la nature, en sont une suite aussi nécessaire que les révolutions du soleil. Dans l'ignorance des liens qui les unissent au système entier de l'univers on les fait dépendre des causes finales, ou du hasard, suivant qu'ils arrivaient et se succédaient avec régularité ou sans ordre apparent. Mais ces causes imaginaires ont été successivement reculées avec les bornes de nos connaissances, et disparaissent entièrement devant la saine philosophie, qui ne voit en elles que l'expression de l'ignorance où nous sommes des véritables causes. LAPLACE, Essai philosophique sur les Probabilités, p. 2.

mind noted by Bacon,* according to which an uniformity observed soon becomes generalized. Simplicity is so gratifying to the mind, that we are impatient of all pertur bations, and huddle them out of sight, inclining to believe that whatever is simple must be truer than what is not. This leads to many precipitate judgments which Experience refutes. For example, nothing can be simpler than the law which declares that acids combine definitely with bases to form salts, and in these combinations the properties of the substances are mutually neutralized. What says Experiment? It says that the combination of an acid with a base does not uniformly, invariably result in this neutralization: sometimes (in what are called the acid salts) the acid properties continue to be manifest; sometimes (in the basic salts) the alkaline properties appear. It says, moreover, that oxides, and even oxides of the same metal, combine with each other, and that acids sometimes combine with neutral substances (e. g. sulphuric acid with chloride of sodium), or neutral substances with each other. It also says that the same substance will act as an acid in one combination, and as a base in another. Thus, simple the law of combination may be, and true as an abstraction, yet the concrete. phenomena present so many diversities as to suggest that the law itself is only an incompletely stated case of some more general law of combination. In a word, the distinction between acids and alkalies vanishes on a close analysis; the terms appear only as the two extremes of a series in which the intermediate terms participate more or less. in the general characters of acid and alkali by analogy of composition or properties, without, however, possessing

*"Intellectus humanus ex proprietate sua facile supponit majorem ordinem et æqualitatem in rebus quam invenit; et cum multa sint in natura monodica [monadica] et plena imparitatis tamen affingit parallela et correspondentia et relativa quæ non sunt."— BACON, Novum Organum, Aph. 45.

the specially distinctive characters of either; just as the different colors we distinguish in the impure spectrum, such as a rainbow, really contain the vibrations of all orders, but in different proportions, the red containing a maximum of red vibrations with a minimum of violet, and so on.

92. The generalization that all phenomena are regulated by Law requires interpretation. We are not to suppose that Law is an objective real acting in phenomena; it is the ideal conception of the phenomena themselves, classified according to their resemblances with other phenomena. The Law of Nature has no more a concrete existence, apart from the changes in the relations of phenomena, than a Genus exists apart from the individuals it comprises. Hence Law means (in mathematical phrase) the function of the phenomena; and the generalization that Chance (i. e. the emancipation of phenomena from Causality) has no place in the system of things is simply the obverse of the previous generalization. So far all is clear; but now observe the consequence. If all events have their law, each event has its law, namely, that under like conditions it will be invariable; and if the events in Nature are complicated and changeable, what is called the simplicity and uniformity of Nature is not what exists and is observed, but what is constructed in Abstraction, letting drop the observed complexities and irregularities. The invariability we find in Nature is what we have put there. Thus a body moved by various impulses, and by several velocities, will describe a curve which geometers show might equally well be described under the action of the single resultant force. Because this curve has a simple expression, we might, without further knowledge, regard the law as simple; yet it is obviously complex.

CHAPTER VI.

THE LAW OF INVARIANTS.*

93. WE have reached the ultimate logical principle which is the expression of the test of Certitude. But the principle of Equivalence (in the terms equated) has only a logical or subjective aspect; we must now see it in its correlative real or objective aspect, as a cosmical law.

A moment's consideration will disclose that facts or events are either the sums of their units or the products of their factors. The difference between an aggregate and a product is that in the first case the component parts are simply grouped together, added; in the second, the constituent elements are blended, multiplied into each other. (Compare RULE IX.) But in every case the phenomenon is what it is in virtue of its determinants. These determinants (causes, conditions) are quantitatively and qualitatively invariant, the same values always co-operating to produce the same result. There must be variable elements for varying phenomena; but each phenomenon in itself, within its own limit, is necessarily the resultant or the emergent † of units and factors that are invariant. Thus the number 10, for instance, may be formed by the addition of 5 to 5, of 3 to 7, of 3 to 5 and

* This term Invariant has no reference to the speculations of a distinguished modern school of mathematicians. It is here used instead of Invariable, to avoid many misconceptions.

✦ On the distinction between resultant and emergent, see PROBLEM V. § 63.

2, of 8 to 2, of 6 to 3 and 1,- so many variable elements, each of which is, however, constituted by invariants, and the sum of product of invariants must likewise be invariant. In Dynamics two forces are identical when, acting for the same time, they move the same mass with the same velocity in the same direction, although these forces. may be different in their proximate origin and accompaniments, the one being a muscular contraction, the other the expansion of an elastic fluid, a third the impulse of a solid. But however variable the visible antecedents may be, the real determinants the co-operant factors are in each case invariant.

94. Here, in passing, note the common fallacy of ascribing the same effect to different causes. (Compare RULE VI. and PROBLEM V. Chap. III.) A close consideration will show that the same effect is everywhere produced, and is only producible, by the same cause, since the product can express only its factors. The attendant circumstances, which perhaps mask the real determinants, lie outside the causal relation; they are not co-operant factors. The weight of a body, for instance, is not determined by its color, form, temperature, etc., but by the quantity, or density, of its molecules, and its relative position in space. Starch, again, is converted into glucose by one cause, and one only, though this determinant may be obtained by a cortège of circumstances which are not conditions of the result, but mere camp-followers, taking no active part in the struggle. Chemists call this determinant the hydratation of starch, that is, the fixing in the starch of one equivalent of water, OH. This fixing may be brought about in various ways, by heat, vegetable diastase, acidulated water, etc.; and if we regard-and usually we do regard these agents as the causes, it will be true to say that different causes have here produced the same effect. But this is the popular