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 course, does not imply that there were not certain elements in Schelling, particularly in the Transcendental Idealism, that are of value in the transition to the later system; but on the whole it is only in Hegel that the whole matter of the Kantian categories has been assimilated and carried to a higher stage. The Hegelian philosophy, in brief, is a system of the categories; and, as it is not intended here to expound that philosophy, it is impossible to give more than a few general and quite external observations as to the Hegelian mode of viewing these elements of thought. With Kant, as has been seen, the categories were still subjective, not as being forms of the individual subject, but as having over against them the world of noumena to which they were inapplicable. Self-consciousness, which was, even with Kant, the nodus or kernel whence the categories sprang, was nothing but a logical centre,—the reality was concealed. There was thus a dualism, to overcome which is the first step in the Hegelian system. The principle, if there is to be one, must be universally applicable, all-comprehensive. Self-consciousness is precisely the principle wanted; it is a unity, an identity, containing in itself a multiplicity. The universal in absolute self-consciousness is just pure thinking, which in systematic evolution is the categories; the particular is the natural or multiform, the external as such; the concrete of both is spirit, or self-consciousness come to itself. The same law that obtains among the categories is found adequate to an explanation of the external thing which had so sadly troubled Kant. The categories themselves are moments of the universal of thought, type forms, or definite aspects which thought assumes; determinations, Bestimmungen, as Hegel most frequently calls them. They evolve by the same law that was found to be the essence of ultimate reality—i.e. of self-consciousness. The complete system is pure thought, the Universal par excellence.

After the Hegelian there can hardly be said to have been a philosophical treatment of the categories in Germany which is not more or less a criticism of that system. It does not seem necessary to mention the unimportant modifications introduced by Kuno Fischer, J. E. Erdmann, or others belonging to the school. In the strongly-opposed philosophy of J. F. Herbart the categories can hardly be said to hold a prominent place. They are, with him, the most general notions which are psychologically formed, and he classifies them as follows:—(1) Thing, either as product of thought or as given in experience; (2) Property, either qualitative or quantitative; (3) Relation; (4) The Negated. Along with these he posits as categories of inner process—(1) Sensation, (2) Cognition, (3) Will, (4) Action. Joh. Fr. L. George (1811–1873), who in the main follows Schleiermacher, draws out a table of categories which shows, in some points, traces of Herbartian influence. His arrangement by enneads, or series of nine, is fanciful, and wanting in inner principle.

The most imposing of more recent attempts at a reconstruction of the categories is that of F. A. Trendelenburg. To him the first principle, or primitive reality, is Motion, which is both real as external movement, and ideal as inner construction. The necessary conditions of Motion are Time and Space, which are both subjective and objective. From this point onwards are developed the mathematical (point, line, &c.) and real (causality, substance, quantity, quality, &c.) categories which appear as involved in the notion of motion. Matter cannot be regarded as a product of motion; it is the condition of motion, we must think something moved. All these categories, “under the presupposition of motion as the first energy of thought, are ideal and subjective relations; as also, under the presupposition of motion as the first energy of Being, real and objective relations.” A serious difficulty presents itself in the next category, that of End (Zweck), which can easily be thought for inner activity, but can hardly be reconciled with real motion. Trendelenburg solves the difficulty only empirically, by pointing to the insufficiency of the merely mechanical to account for the organic. The consideration of Modality effects the transition to the forms of logical thought. On the whole, Trendelenburg’s unique fact of motion seems rather a blunder. There is much more involved than he is willing to allow, and motion per se is by no means adequate to self-consciousness. His theory has found little favour.

Hermann Ulrici works out a system of the categories from a psychological or logical point of view. To him the fundamental fact of philosophy is the distinguishing activity (unterscheidende Tätigkeit) of thought. Thought is only possible by distinction, difference. The fixed points in the relations of objects upon which this activity turns are the categories, which may be called the forms or laws of thought. They are the aspects of things, notions under which things must be brought, in order to become objects of thought. They are thus the most general predicates or heads of predicates. The categories cannot be completely gathered from experience, nor can they be evolved a priori; but, by attending to the general relations of thought and its purely indefinite matter, and examining what we must predicate in order to know Being, we may attain to a satisfactory list. Such a list is given in great detail in the System der Logik (1852), and in briefer, preciser form in the Compendium der Logik (2nd ed., 1872); it is in many points well deserving of attention.

The definition of the categories by the able French logician Charles Bernard Renouvier in some respects resembles that of Ulrici. To him the primitive fact is Relation, of which all the categories are but forms. “The categories,” he says, “are the primary and irreducible laws of knowledge, the fundamental relations which determine its form and regulate its movements.” His table and his criticism of the Kantian theory are both of interest. The criticism of Kant's categories by Cousin and his own attempted classification are of no importance. Of little more value is the elaborate table drawn out by Sir W. Hamilton. The generalized category of the Conditioned has but little meaning, and the subordinate categories evolve themselves by no, principle, but are arranged after a formal and quite arbitrary manner. They are never brought into connexion with thought itself, nor could they be shown to spring from its nature and relations. J. S. Mill presented, “as a substitute for the abortive classification of Existences, termed the categories of Aristotle,” the following as an enumeration of all nameable things:—(1) Feelings, or states of consciousness; (2) The minds which experience these feelings; (3) Bodies, or external objects which excite certain of those feelings; (4) Successions and co-existences, likenesses and unlikenesses, between feelings or states of consciousness. This classification proceeds on a quite peculiar view of the categories, and is here presented only for the sake of completeness.

By modern psychologists the subject has been closely investigated. Professor G. F. Stout (Manual of Psychology, vol. ii. pp. 312 foll.) defines categories as “forms of cognitive consciousness, universal principles or relations presupposed either in all cognition or in all cognition of a certain kind.” He then treats External (or Physical) Reality, Space, Time, Causality and “Thinghood” from the standpoint of the perceptual consciousness; showing in what sense the categories of causality, substance and the rest exist in the sphere of perception. As contrasted with the ideational, the perceptual consciousness is concerned with practice. Perception tells the child of things as separate entities, not in their ultimate relations as parts of a coherent whole. G. T. Ladd (Psychology Descriptive and Explanatory, ch. xxi., on “Space, Time and Causality”) defines the categories from the psychological standpoint as “those highly abstract conceptions which the mind frames by reflection upon its own most general modes of behaviour. They are our own notions resulting from co-operation of imagination and judgment, concerning the ultimate and unanalyzable forms of our own existence and development.” In other words, the categories are highly abstract, have no content, and are realized as a kind of thinking which has for its object all the other mental processes.

.—Besides those quoted above, see Eduard v. Hartmann, Kategorienlehre (Leipzig, 1896), and “Begriff der Kategorialfunktion,” in ''Zeitschr. f. Philos. und phil. Krit.'' cxv. (1899), pp. 9-19; E. König in the same periodical cxiii. (1889), pp. 232-279, and cxiv. (1899), pp. 78-105; F. A. Trendelenburg, Geschichte der Kategorienlehre (1846); P. Ragnisco, Storia critica delle categorie (2 vols., Florence, 1871); W. Windelband Vom System der Kategorien (Tübingen, 1900); R. Eisler, Wörterbuch der philosophischen Begriffe (Berlin, 1899), pp. 400-409; S. Joda, Studio critico su le categorie (Naples, 1881); H. Vaihinger, Die transcendentale Deduktion der Kategorien (Halle, 1902); H. W. B. Joseph, Introduction to Logic (Oxford, 1906), ch. iii.; F. H. Bradley, Principles of Logic (1883); B. Bosanquet’s Knowledge and Reality (1885, 2nd ed. 1892); histories of philosophy. For further authorities see works quoted under and, and in J. M. Baldwin’s ''Dict. Philos. Psych.'' vol. iii. pt. 2, p. 685.

 CATENARY (from Lat. catena, a chain), in mathematics, the curve assumed by a uniform chain or string hanging freely between two supports. It was investigated by Galileo, who erroneously determined it to be a parabola; Jungius detected Galileo’s error, but the true form was not discovered until 1691, when James Bernoulli published it as a problem in the Acta Eruditorum. Bernoulli also considered the cases when (1) the chain was of variable density, (2) extensible, (3) acted upon at each point by a force directed to a fixed centre. These curves attracted much attention and were discussed by John Bernoulli, Leibnitz, Huygens, David Gregory and others.