stratum resemble each other the more as totalities the larger they are, however manifold their personal variations may be; on the other hand, special minor centers of common conversation, temper, interest, which, however, incessantly interchange their members. There thus occurs that incessant variation of attachment and detachment in the large society which, according to the nature of the persons concerned, affects one now as the most intolerable superficiality and again as a rhythmic play of the highest æsthetic charm. The ball, with the modern form of dance, exhibits this formal sociological type in a quite distinct example: a momentary and quite wonderfully close relationship of each single pair, modified into a quite new formation through constant changes among the pairs; this physical contiguity between persons quite strange to each other, on the one side, made possible by the fact that they are all guests of a single host who, however loose the relationship to himself may be, affords a certain reciprocal security and legitimation; on the other hand, through the unpersonal and, so to speak, anonymous character of the relationship which the size of the society and the formality of behavior joined with it, produce. Evidently these traits of the large "society" which the ball presents at once in a sublimated form and perhaps in caricature, are attached to a definite minimum number of participants; and we may often make the most interesting observation that an intimate circle of a few persons attains the character of a "society" through the appearance of a single additional person.

In a case which, to be sure, concerns a much less complicated quality, the number which produces a definite sociological unitary structure is somewhat more firmly fixed. The patriarchal domestic family numbers, in the most various regions, always twenty to thirty heads, and that, too, under quite dissimilar economic conditions, so that these cannot, or at least cannot exclusively, occasion the numerical equality. It is rather probable that the inner reciprocities which constitute the special structure of the domestic family produce the proportions of closeness and extension demanded for this structure only within just these boundaries. The patriarchal family has been every

where characterized by great intimacy and solidarity, which had its center in the paterfamilias, through the guardianship which the latter exercised, both in the interest of the whole and in his own egoistic interest, over the concerns of each individual. Hence the upper boundary was determined: this sort of dependence and of control appears able to comprehend, at the corresponding stage of psychological development, no greater collection of elements. The lower boundary, on the other hand, is determined by the fact that a group thus dependent upon itself for its selfsatisfaction and its maintenance must develop certain collective psychical facts, which in turn usually arise only above a certain numerical limit. Such facts are, for example, resolution for offense and defense, confidence on the part of each that he will at every moment find the necessary support and reinforcement; more than all religious consensus, whose exaltation and spirituality can raise itself above the individual and the individual above himself only from the commingling of many contributions, with reciprocal effacement of their individual peculiar character. The number mentioned has perhaps indicated the scope approximately established by experience above and below which the group could not go if it were to develop the traits of the patriarchal house family. It appears as though, with increasing individualization, before this stage of culture, those intimacies were possible only within a smaller number of persons; the phenomena, on the other hand, which looked to the size of the family at once demanded an ever-growing circle. The needs from above and below which in that stage of culture realized themselves with this numerical material have become differentiated; one portion demands a smaller, another a larger number, so that later no structure is any longer in existence which can satisfy both sides of the demand in the same unified way as was the case with the patriarchal family.

Apart from such singular cases, all questions of the sort whose type is the numerical requirements for a "society" have a sophistical tone-how many soldiers make an army, how many members are necessary to constitute a political party, how many participants make an outbreak. They appear to rehearse the

classical riddle: How many grains of wheat make a heap? For since one, two, three, four grains do not do it, but a thousand certainly, there must accordingly be between these two numbers a boundary, at which the addition of a single grain makes those previously present into a "heap." If one, however, makes the trial of further enumeration, it appears that no one can announce discovery of this boundary. The logical ground of these difficulties is found in the fact that a quantitative series is given which, on account of the relative insignificance of each single element, seems to be a continuous and uniformly ascending series, and that this from a certain point on must permit the application of a qualitatively new idea, sharply set off from the idea previously applied. This is obviously a contradictory demand: by virtue of its very idea, the continuous cannot justify purely of itself a sudden break and change. The sociological difficulty has now a further complication aside from that in the ancient sophistry, for by the "heap" of grains we understand either a piling-up, and then one is logically justified in this use of terms so soon as only one layer appears above the lowest layer; or, only a quantity is designated by the term. In this case it is quite unjustifiable to demand of an idea like "heap," which in its very essence is quite variable and undetermined, that it should lend itself to application to perfectly defined and unequivocally bounded

realities.'

In those sociological cases, however, specifically new aggregate phenomena appear, when quantity increases, which are not present pro rata in the case of smaller numbers. A political party has qualitatively another significance from that of a small clique. A few curious persons standing together betray different traits from those of a mob (Auflauf), etc. The indeterminateness which attaches to these ideas from the impossibility of fixing numerically the corresponding quantities may perhaps be

'Still more evident is this mistake in the negative direction, in the case of the question: How many hairs must one lose before one may be called bald? If we take this latter idea seriously, it applies only to him who has no hair at all. If we apply it, however, to any case in which there is possession of hair, we thereby surrender the unequivocal severity of the idea, and we may not wonder that we possess no objectively precise criterion of its application, since we have put such application out of our power.

emphasized in the following way. This variation affects evidently only certain intermediate quantities. Certain lower numbers surely do not yet constitute the collectivities in question. Certain quite higher numbers constitute them without doubt. But even those numerically small structures have, however, sociological qualities which are characteristic: the meeting which falls short of being the "society," the troop of soldiers which does not constitute an army, the co-operating vagabonds who are not yet a "band." Since the other qualities, which are quite as little doubtful in the case of the great society, are in contrast with these, we may indicate the character of those that are numerically intermediate as composed of both, so that each makes itself in a rudimentary way perceptible in particular characteristics-now appears, now disappears or becomes latent. Since, then, such structures, lying in the intermediate numerical zone, have also a share objectively in the decided character of the structure below and above, or at least have such share partially and intermittently, the subjective uncertainty in determining to which of them they belong is to be explained accordingly. The point, then, is not that in a sociologically quality less structure suddenly, like the crystal in the solution, a quite definite sociological constellation forms, without our being able to designate the precise moment of this transformation; but rather that two diverse formations, each consisting of a collection of traits and capable of being arranged in a qualitative scale, under certain quantitative conditions meet in a social structure, and in various degrees divide it between themselves; so that the question to which of the two they belong is not essentially one that suffers from the inherent epistemological difficulties of continuous series, but it is simply a question whose content is fallaciously proposed.'

I More precisely, however, the situation is rather this: To every definite number of elements there corresponds, in accordance with the purpose and spirit of their combination, a sociological form, an organization, firmness of texture, relation of the whole to the parts, etc., which experiences a modification, however immeasurably small and indeterminate, with every added or subtracted element. Since, however, we do not possess a special expression for each of these endlessly numerous sociological conditions, even in those cases in which their distinctive character is observable, there

These explanations have thus had to do with social formations which, to be sure, depended upon the number of the co-working elements, while at the same time our knowledge was insufficient so to formulate this dependence that we could draw from distinct numbers their sociological consequences. Meanwhile this latter is not absolutely excluded, if we turn our attention by preference to sufficiently simple structures. If we begin with the lower boundary of the numerical series, arithmetically defined magnitudes appear as unequivocal preconditions of characteristic sociological formations.

The numerically simplest formations which can at all be designated as social reciprocities appear to occur in the case of reactions between two elements. Yet there is a structure still simpler in external appearance, which belongs in sociological categories, namely, however paradoxical and essentially contradictory it seems, the isolated individual man. As a matter of fact, the processes which produce formations in the case of a duality of elements are often simpler than those necessary for the sociological characterization of the integer. In the case of these latter we have to do chiefly with two here pertinent often remains no alternative but to think the situation as composed of two conditions; the one more and the other less conspicuous. At all events we have to do in such a case as little with a composition as, for example, in the case of the so-called mixed feelings of friendship and love, or hate and contempt, or pleasure and pain. In these cases, at least in numberless instances, a quite unified condition of feeling exists, for which we merely have no immediate concept, and which we consequently, by means of the synthesis and reciprocal limitation of two other concepts, rather circumscribe than describe; here, as in other instances, the proper unity of the existing is not within our comprehension, but we must resolve it into a duality of elements, neither of which quite covers it, in order to think it as proceeding from the interweaving of the two. This is, however, merely a conceptual analysis, possible only after the fact, which does not trace the actual genetic process, the proper being of that unity. Where therefore the stereotyped concepts for social unities-meeting and society, troop and army, clique and party, a few and a band, personal following and school, crowd and popular uprising, etc., etc.—do not find an exact application, because the human material seems to be insufficient for the one, and more than enough for the other, there yet exists a sociological mold, precisely as unified and just as specifically corresponding to the numerical limitation as in those more definite' cases. The fact merely is that the lack of a special concept for these innumerable shadings compels us to denote their qualities as a mixture of those forms which correspond to the numerically inferior and to the numerically higher structures.