controlled splendour of Thought-activity, which has ever been the only 'Light of the World,' some more true, or at least, less untrue tidings of the incommensurable reality of Things?

SUBSECTION II.

A Classification of the Sciences, and the Arts.

1. The remarks with which we have just concluded our statement of the Proximate Principles of this New Method may already have suggested that the most important illustration, as indeed the immediate result of the application, of the most characteristic of these principles, will be a classification of the Sciences. The subject of our enquiry is Causation. The distinguishing principle of the method of our enquiry demands a procedure at once progressive and systematic in our investigation of the relations of things. Hence, there arises a system of conceptions, which are drawn, in the first instance, from investigation of the simplest relations of things; these are then defined and systematised by being brought into relation with other conceptions; and-these all being held only as provisional generalisations or hypotheses these conceptions are then submitted to deductive verification, and, according to the results of that, rejected or retained as truly correlative. But such conceptions will define the various departments of a System of Knowledges. A Classification, therefore, of the Sciences, or Systematisation of Knowledges will thus, evidently, be the outward form, as it were, or embodiment of the principles of our New

Method. And hence, in order to a clear comprehension of these principles, it will be necessary for me to give the outlines, at least, of such an embodiment. What the steps, however, were of this classification, how various the changes in the course of it, and how numerous the tabular reconstructions in the attempt to bring the antitheses of Thought into accordance with the relations of Things, it would be out of place here to note. Nor will I make any further preliminary remark than that, to be in accordance with the general aim of the method stated in the above-enunciated principles, the classes of the sciences should correspond, both in matter and in form, with the laws which are their respective contents. Both in matter and in form. For a law, in one point of view, is an objective relation of Things, and, in another aspect, a subjective mode of Thought. Hence, the classes of the sciences, as distinguished by this method, should correspond, at once, with the general categories of Things, and with the fundamental processes of Thought. The aim, therefore, of our systematisation will be to classify Things by their real relations, and Knowledges by their true methods. And if this aim should be in any degree realised, our Classification may have some claim, perhaps, to that highest of all merits which would be implied in the application to it of the epithet natural.1

The phrase Natural Classification seems most peculiarly appropriate to such arrangements as correspond in the groups which they form to the spontaneous tendencies of the mind, by placing together the objects most similar in their general aspect; in opposition to those technical systems which, arranging things according to their agreement in some circum

2. Now, proceeding on the Method, the principles of which have been just stated, hence, forming our general conceptions from investigation of the actual relations of Things, and beginning with the simplest of these relations; we shall, I think, be led to consider formal relations of Position, or quantitative relations, as the true starting-point, both of our investigation of Things, and of our systematisation of Knowledges. With the mathematical sciences, therefore, we begin. But now, how are these to be classified? How are the quantitative relations, the subject-matter of Mathematic, to be distinguished and connected? What are the various kinds of formal relations of Position? The answer to these questions is to be found in the investigation of the history, present development, and tendencies of the mathematical sciences. But here I can only remark that, since Descartes' great discovery of a general method of reducing conceptions of Position to conceptions of Magnitude and Number,1 geometry has not only tended more and more to be absorbed in analysis, or algebra; but our conception of the very basis of it has been modified through recent speculations on the possible curvature of our three

stance arbitrarily selected, often throw into the same group objects which, in the general aggregate of their properties, present no resemblance, and into different and remote groups, others which have the closest similarity.'-Mill, System of Logic, vol. II. p. 265. Compare Cuvier, Règne animal, Introd. See also Ueberweg, System of Logic, § 63, Division.

1 This mathematical discovery of Descartes' will, on reflection, be seen to have a profound connection with the general change in philosophic conception indicated by his famous axiom Cogito, ergo sum. For Thought is sequence, and Matter, coexistence. And to reduce conceptions of Position to conceptions of Number is to reduce conceptions of Coexistence to conceptions of Sequence.

dimensioned Space.1 Since Descartes, then, the conception of Position has become generally expressible in terms of Number. And, by our Second Principle, developing our conception of Position, or of Number, the conception to which it may be reduced, we find that it may be regarded either as discontinuous, or continuous.2 May not, then, the sciences of Mathematic be distinguished as sciences, first, of discontinuous, and secondly, of continuous Position? But again, our Second Principle, as one of integration, as well as of differentiation, suggests a third class of mathematical truths integrating the conceptions of the two previous classes in a science of ordered Position.3 The first class might be named Arithmetic in the most .general sense of the term, and as including algebra in its ordinary signification; the second class, Algebraic,

1 Euclid's solid space is a homoloid. And it is asked why this solid should be under a disability to which the line and the plane are not subjected-why should it not, as well as the line and the plane, be capable of curvature? See Riemann On the Hypotheses, and Helmholtz On the Facts upon which Geometry is based; the former, in the Abhandl. der Königl. Gesellsch. d. Wissensch. zu Göttingen; the latter, in the Nachrichten of the same, June 3, 1868.

2 The subject-matter of arithmetic, or of algebra (commonly so called), is discontinuous number. . . . . Infinitesimal calculus, on the contrary, considers number in its aspect of continuous growth.'-Price, Infinitesimal Calculus, vol. 1. pp. 16–17.

3 Dr. Ingleby, to whom, in the beginning of 1871, I communicated these conceptions of discontinuity, continuity, and order as those on which I proposed to classify the Mathematical Sciences, greatly encouraged me by remarking that the late Sir W. R. Hamilton had, in conversation with him some years before his death, defined mathematics as the Science of arrangement in Time, Space, and Order.' Compare the classification of Hegel, Encyclopädie (Werke, b. VII. a.), and the division Quantität of Die Lehre von Seyn Logik (Werke, b. III.); that of Comte, Philosophie positive, t. 1. leç. iii.; that of Ampère, Philosophie des Sciences, t. 1. pp. 32-54; and that of Spencer, Classification of the Sciences, p. 15.

4 Prof. De Morgan had 'no doubt that Algebra got its Arabic name al

if its subject-matter is considered as Position, or Number in its continuous aspect; and the third class might be named Tactic.2

3. Having thus exhausted the conception of Position in its three general forms of discontinuity, continuity, and order, we proceed to the differentiation of this conception. Motion, and its systematic or causal relations, suggests itself as the correlate of Position, and its sequential or formal relations. Whether this conception is thus truly differentiated or not must, by our principle of verification, be decided by investigation of the actual relations jebr e al mokābala, restoration and reduction, from the restoration of the term which completes the Square, and reduction of the equation by extracting the square root-the solution of a quadratic equation being the prominent part of Arabian Algebra. Trigonometry and Double Algebra, p. 98, n. In his Elements of Algebra, p. xxxvii., he distinguishes an arithmetical problem as one in which numbers are given, and certain operations; and an algebraical problem as one in which numbers are either given or supposed to be given, and a question is asked of which it is not at once perceptible what operations will furnish the answer. Comte includes in Arithmetic, tout ce qui a pour objet l'évaluation des fonctions.' (Philosophie positive, t. I. p. 184.) Compare Price, Infinitesimal Calculus, as above cited, and Peacock, Algebra, Arithmetical and Symbolical, vol. I. ch. i. Compare also with the latter De Morgan, Trigonometry, book II. ch. ii. On Symbolic Algebra.

1 Lagrange defined Algebra as ‘le Calcul des Fonctions;' and citing this definition, Sir W. R. Hamilton says: "It is not easy to conceive a clearer or juster idea of a function in this science, than by regarding its essence as consisting in a law connecting change with change.'-Theory of Conjugate Functions, Trans. Royal Irish Acad. vol. XVII. p. 290. Note also that Trigonometry, or to speak more properly Goniometry, (Peacock, Algebra, vol. II. p. v.), as a branch of algebra, is defined by De Morgan, as 'the science of continually undulating magnitude.'-Trigonometry, p. 1; but compare p. 20, note.

This term was first invented by Dr. Sylvester to denote a certain special department of algebraical research. And whether it can now be conveniently used with such a meaning as that given to it in the text must depend on his approval, and that of Professor Cayley and the other eminent mathematicians by whom the term has, in Dr. Sylvester's sense of it, been employed. But no more convenient term suggests itself to me.