Convergence

“ When , the terms of series (G) will, after the first, be greater than the corresponding terms of the harmonic series; hence (G) is divergent.
139. Cauchy's ratio test for convergence. Let
be a series of positive termes to be tested.
Divide any general term by the one that immediately precedes it; i.e. form the test ratio .
As increases without limit, let .
I. When . By the definition of a limit (§13) we can choose n so large, say , that when the ratio shall differ from by as little as we please, and therefore be less than a proper fraction . Hence
and so on. ”

“ Social phaenomena, like all others, present two aspects, the statical, and the dynamical; the phaenomena of equilibrium, and those of motion. The statical aspect is that of the laws of social existence, considered abstractedly from progress, and confined to what is common to the progressive and the stationary state. The dynamical aspect is that of social progress. The statics of society is the study of the conditions of existence and permanence of the social state. The dynamics studies the laws of its evolution. The first is the theory of the consensus, or interdependence of social phaenomena. ”

“ The only obvious respect in which all kinds of Progress are alike, is, that they are modes of change; and hence, in some characteristic of changes in general, the desired solution will probably be found. We may suspect à priori that in some law of change lies the explanation of this universal transformation of the homogeneous into the heterogeneous. 32 Thus much premised, we pass at once to the statement of the law, which is this:—Every active force produces more than one change—every cause produces more than one effect. ”

“ But the change, though steadily and rapidly progressive, has nevertheless been gradual; and, like those who drift down the stream of a deep and smooth river, we are not aware of the progress we have made until we fix our eye on the now distant point from which we have been drifted. ”

“ Whether it be in the development of the Earth, in the development of Life upon its surface, in the development of Society, of Government, of Manufactures, of Commerce, of Language, Literature, Science, Art, this same evolution of the simple into the complex, through successive differentiations, holds throughout. From the earliest traceable cosmical changes down to the latest results of civilization, we shall find that the transformation of the homogeneous into the heterogeneous, is that in which Progress essentially consists. ”

“ Just as we have become acquainted with phenomena of progress, quite independent of divergence, so we shall, on the other hand, very often meet with divergencies which285 are not perfecting, but which are rather the contrary, that is retrogressions or degenerations. ”

“ But that it would take us too far out of our track, we should prelude this inquiry by illustrating at some length a certain general law of progress;—the law that alike in occupations, sciences, arts, the divisions that had a common root, but by continual divergence have become distinct, and are now being separately developed, are not truly independent, but severally act and react on each other to their mutual advancement. ”

“ But that all that we call "being" is motion: and that all motion is the expression, not of equilibrium, but of equilibrating, or of equilibrium unattained: that life-motions are expressions of equilibrium unattained: that all thought relates to the unattained: that to have what is called being in our quasi-state, is not to be in the positive sense, or is to be intermediate to Equilibrium and Inequilibrium. ”

“ Regarding the functions as constituting a moving equilibrium, we may say that divergence of any function in the direction of increase, causes the functions with which it is bound up to diverge in the same direction ”

“ Surely, then, in such complex and slowly-evolving movements as those of a nation's life, all the smaller and greater rhythms of which fall within certain general directions, it is impossible that such general directions can be traced by looking at stages that are close together—it is impossible that the effect wrought on any general, direction, by some additional force, can be truly computed from observations extending over but a few years, or but a few generations. ”

“ Ye’re all agreed that societies change—not always and everywhere—but on the whole and in the long run. Now, with all deference, I would beg t’ observe that we have got to examine the nature of changes before we have a warrant to call them progress, which word is supposed to include a bettering, though I apprehend it to be ill-chosen for that purpose, since mere motion onward may carry us to a bog or a precipice. ”

“ Of course we are: but there is a universal balance throughout nature, and everything finds its level. There is order, when there appears disorder—and no stream runs in one direction, without a counter stream, to restore the equilibrium. ”

“ The great division of history into phases by Turgot and Comte first affirmed this law in its outlines by asserting the unity of progress, for a mere phase interrupts no growth, and nature shows innumerable such phases. ”

“ But, the question may be asked, Granting all this, may not the activity at some time come to an end? May not an equilibrium be finally reached—a state in which, activity having played its part, becoming passes into being, a changeless goal of all preceding change? ”

“ In the higher groups unequal form-divergence may be produced by the transforming impulses affecting parts of unequal physiological and morphological value, or by their influencing parts of equal value in different degrees. ”

“ It is a corollary that when the environment is changed, the equilibrium of functions is disturbed, and there must follow one of two results—either the equilibrium is overthrown or it is re-adjusted: there is a re-equilibration. ”

“ The law that the divergence in form always corresponds exactly with the divergence in the conditions of life212 has accordingly received confirmation in all cases where we have been able to pronounce judgment. ”

“ And thus the ultimate state, forming the limit towards which Evolution carries us, is one in which the kinds and quantities of mental energy daily generated and transformed into motions, are equivalent to, or in equilibrium with, the various orders and degrees of surrounding forces which antagonize such motions. ”

“ But without soaring to the stars, the moment that this theory of progression, which seems so natural to us amid an advancing civilization, looks around the world, it comes against an enormous fact—the fixed, petrified civilizations. ”

“ And the highest form of evolution is reached when there is a harmony between concentration and differentiation, a harmony which Spencer calls equilibration and which he defines as a moving equilibrium. ”

“ Indeed, as this observation suggests, it will be well, before tracing the mode in which exact science finally emerges from the merely approximate judgments of the senses, and showing the non-serial evolution of its divisions, to note the non-serial character of those preliminary processes of which all after development is a continuation. On re-considering them it will be seen that not only are they divergent growths from a common root,—not only are they simultaneous in their progress ”

“ By this mutual transfusion an equilibrium is preserved, like that which similar processes maintain in the natural world, and while a healthy, popular feeling circulates through the Aristocracy, a sense of their own station in the scale elevates the People. ”

“ We may, in fact, characterize the situation generally as the intermediate plane between the universal world-condition still in a state of equilibrium, and the concrete action unfolded in all its tendency to movement and reaction, a position which gives to it the characteristics of both extremes, and enables us to pass over from the one to the other. ”

“ Until this happens, the isolation of single societies must be repeatedly broken in upon and the process of equilibration disturbed by the introduction of new elements to which adjustment must take place; the new adjustment being in the sense of progress towards a higher system of equilibrium, that is, one of more elements, and the whole process constituting a continual progress in the direction of a full stability of Life upon the earth. ”

“ If we should conceive, for example, that the human mind, with its different possibilities of equilibrium, might be like a many-sided solid with different surfaces on which it could lie flat, we might liken mental revolutions to the spatial revolutions of such a body. ”

“ But equilibrium and stability are never perfect. Permanent social equilibrium would be social stagnation and decay. Therefore, we must study society also in its onward movement—the equilibrium ever disturbed, only to be readjusted on a higher plane with more and more complexly inter-related parts. This is dynamics—social progress. It is evolution. ”

“ For considering its individuals as atoms, the growth of society is “an integration of matter and concomitant dissipation of motion; during which the matter passes from an indefinite, incoherent homogeneity to a definite, coherent heterogeneity, and during which the retained motion undergoes a parallel transformation.”60 And thus an analogy may be drawn between the life of a society and the life of a solar system upon the nebular hypothesis. ”

“ That a constant, but almost imperceptible change is hourly taking place in the earth's surface, appears to be established; and independent of the extraordinary bouleversements, which have at intervals convulsed our globe, this gradual revolution has produced, and will produce again, a total alteration in the face of nature. ”

“ The new set of forces, compounded with all the antecedent sets of forces, can but inappreciably modify that moving equilibrium of functions which all these antecedent sets of forces have established. Though there may result a considerable perturbation of certain functions—a considerable divergence from their ordinary rhythms—yet the general centre of equilibrium cannot be sensibly changed. ”

“ When we do meet with a clear and unmistakable series of such progressive magnitudes or ratios, manifesting themselves in a progressive series of “allied” forms, then we have the phenomenon of “orthogenesis.” ”

“ Tendency to equilibrium of force, and to permanency of form then, are the characters of that portion of the universe which does not live—the domain of the chemist and physicist.
Tendency to disturb existing equilibrium,—to take on forms which succeed one another in definite cycles, is the character of the living world. ”

“ The definite combination of heterogeneous changes, both simultaneous and successive, is a formula which fails to call up an adequate conception. And it fails from omitting the most distinctive peculiarity—the peculiarity of which we have the most familiar experience, and with which our notion of Life is, more than with any other, associated. ”

“ But the cataclysm once accomplished, a new uniformity must have supervened, and evolution proceeded from that point continuously from the simple to the more heterogeneous, in accordance with strict dynamical law. ”

“ The equilibrium which is due to such passive properties is thus widely distinguished from that caused by the balance of the active tendencies of the system, where an external influence, or a change in the initial state, infinitesimal in amount, is sufficient to produce change either in the positive or negative direction. ”

“ Material progress, as commonly understood, consists of those three elements or directions of progression, in all of which the progressive nations have for some time past been advancing, though in different degrees. ”

“ Every more circumscribed example of development according to a plan, this view regards as a single case in which, out of the general course of nature, and fully accounted for by it, single groups of its elements are arranged into a totality whose cohering unity consists only in the reciprocal actions of the combined elements themselves. ”

“ Yet these movements are in many ways in conflict with one another, and they naturally tend to bring one another to a uniformity—that is, into a state in which one movement is as little obstructive to the other as possible. ”

“ The indefiniteness of form, the discontinuity of the parts, the mobility of the parts, and the universal sensitiveness, are not only peculiarities of the social organism which have to be stated with considerable qualifications; but they are peculiarities to which the inferior classes of animals present approximations. Thus we find but little to conflict with the all-important analogies. That societies slowly augment in mass; that they progress in complexity of structure ”

“ This rhythm of evolution and dissolution, completing itself during short periods in small aggregates, and in the vast aggregates distributed through space completing itself in periods which are immeasurable by human thought, is, so far as we can see, universal and eternal—each alternating phase of the process predominating now in this region of space and now in that, as local conditions determine. ”

“ If we omit the phenomena of Dissolution, which are changes from the special to the general, all changes which matter undergoes are from the general to the special—are changes involving a decreasing generality in the united groups of attributes. This is the progress of things. The progress of thought, is not only in the same direction, but also in the opposite direction. The investigation of Nature discloses an increasing number of specialities ”