can we divest ourselves of our intuitions, and form a mental picture of what the universe is to different intuitions. We can indeed symbolically construct a space of two divisions, simply by employing only the symbols of two, and dropping that of the third; as we can construct a geometric figure without attending to its solidity or its color. By such artifices we can conceive, and reason about, the world of the blind; but we cannot picture it. Waiving this point, however, let us note how widely different is the case with a Space of four dimensions. It is obviously impossible to imagine this fourth, which, never having been present to Sense, cannot be revived in Imagination. The comparatively easy resource of dropping one part of our sensible experience, and attending only to the other two, is altogether different from the task of adding an entirely new sensible basis. A fourth dimension, then, must always remain an artifice, which cannot be interpreted in terms of sensible experience. We cannot imagine it, we cannot believe in it as a reality. To accept it on the faith of analytical operations, and to suppose that a manipulation of symbols without regard to sensible experience can lead to anything more than symbolical results, is like supposing that the imaginary creations of poets have a real existence in the sensible world. Genii compressible into bottles, and expansible into giants, can be written about and pictured, but they are not possible realities, which any fisherman may pull up in his net.

B.

LAGRANGE AND HEGEL: THE SPECULATIVE METHOD.

OUR exposition of the limitations to which Deduction is confined carries with it a condemnation of the Method so dear to Metempirics. To complete the lesson, however, we should disengage the real efficiency of a procedure which, although often a failure, is sometimes a success; and to do this we must find out wherein the failure and the success will lie. Two great thinkers, Lagrange and Hegel, may profitably be contrasted as examples of the fertile and infertile employment of the Deductive Method.

In that wonderful achievement, the Mécanique Analytique, Lagrange proposed to himself the novel aim of "reducing the theory

of Mechanics, and the art of resolving its problems to general formulas, the simple development of which gives all the equations necessary for the solution of each problem." He proposed another aim, which was that of "uniting and presenting under one point of view the different principles which had been found to facilitate the solutions of the questions, showing their connection and mutual dependence, enabling us to judge of their correctness and their range." The single principle to which all the others were assigned as developments was the principle of Virtual Velocities. In the opinion of Laplace, this was to render the science perfect. “Il a réduit la recherche du mouvement d'un système quelconque de corps, à l'intégration des équations différentielles. Alors l'objet de la Mécanique est rempli, et c'est à l'analyse pure à achever la solution des problèmes." *

Hegel's aim was to reduce the theory of the Universe, and the solutions of various problems, to a single principle, namely, the dialectical movement of contradiction, in which one idea successively evolved another by union with its opposite. Being and its opposite Non-Being passed from their abstractness into the concreteness of reality, i. e. Becoming. Hegel brought the multiplicity of the Universe under this one rubric, as Lagrange had brought the multiplicity of Motion under his one rubric. The evolution was deductively expounded. Nor can it be said that Hegel's principle is more abstract, and his treatment more analytical, than Lagrange's. If his attempt was pure Metempirics, the attempt of Lagrange was pure Mathematics. If Hegel rejected the complexities of concrete perception and constructed the universe out of conceptions (Begriffe), Lagrange expressed the elementary dynamical relation in terms of the corresponding relations of pure quantities, and from the equation thus obtained deduced his final equations by simple algebra. Thus, although certain quantities which express the physical connections necessarily appear in the equations of motion of the component parts of a system, the method of Lagrange eliminates these quantities from the final equations, and retains simply the algebraical quantities. Nay, so resolute is he to keep to this abstraction, that he declines to call in the aid even of diagrams; fixing attention solely on the symbols, he banishes the ideas of velocity, momentum, and energy, after they have once for all been condensed in the symbols. Strange as this procedure may appear to those who have not reflected on the ideal constructions of Science, it is but an extension

* LAPLACE, Système du Monde, I. 348.

of the principle of Analysis. Science deals primarily with abstractions. All the complexities of concretes are got rid of (when once their abstract values have been ascertained); and thus, in lieu of a mill-stream with its varied banks, "the dark round of the dripping wheel," and the complicated internal mechanism of the mill, Science substitutes abstract numbers: in it the reals disappear and give place to foot-pounds. So in dealing with the diffusion of gases, instead of attempting to follow the real process, the chemist, knowing that the diffusiveness depends on the relative densities of the gases, takes the square root of the number which represents the specific gravity, divides one by this number, and in the fraction thus obtained gets the diffusiveness. Hegel saw clearly enough the triviality of the common objection that Philosophy "deals only with abstractions"; and the common fallacy that therefore it deals only with empty generalities. Philosophy, as he says, "moves only in the region of Thought, and therefore its contents are abstractions; but this is only as respects the form; in its elements Philosophy is concrete." * I think he too often failed steadily to keep the concrete reality in view; but he was assuredly correct in defining Philosophy as the thoughtful contemplation of things, die denkende Betrachtung der Gegenstände; where he erred was in substituting the movement in thoughts as equivalent to the movement in things,

operating on abstract symbols without regard to their concrete reals; a substitution which is perfectly legitimate when the symbols are the rational equivalents of reals, but wholly deceptive when this equivalence is not demonstrable.

It is because Hegel's Method only involves operation on symbols, and not the verification of their equivalence with reals (in this resembling the procedure of all Metempirics), that it conducts him to results flagrantly at variance with some of the best-ascertained truths of Science, and never in any single instance, I believe, conducts him to results which enlarge the store of positive knowledge, out of the purely logical region. Science owes nothing to Hegel's Method, but, on the contrary, has often been seriously retarded by it; whereas Science has been enriched by Lagrange. Hegel has with astonishing ingenuity and consistency ranged the Universe under his one rubric, classifying its phenomena into a system. But the reason why his classification has not the power manifested by Lagrange's is not that he embraces the Universe, Lagrange only embracing Dynamics, but that his logic is uncontrolled by Veri

* HEGEL, Geschichte der Philos., I. 37.

fication. The defect is not simply in "constructing the universe out of conceptions,” since in Philosophy the universe must take this abstract form; the defect lies elsewhere, in constructing the universe out of conceptions which are not the rational equivalents of perceptions. Every reader who has attentively followed the exposition I have given of the process by which rational equivalents are obtained will seize my meaning. Let me, however, illustrate it once more. By rigorous reasoning the principles of Imaginary Geometry prove that two parallel lines would finally meet, and that a line produced would return upon itself. But this Geometry has no methods by which to prove that such lines exist, or that a space of constant curvature is sensible in our Cosmos; and in the absence of such proof we naturally rely on the Geometry which assures us that parallel lines do not and cannot meet in our Cosmos. Were the deductions of Hegel equally rigorous, his Method would still be wholly incompetent to prove that they represented the real order of phenomena, as their rational equivalents, in the same sense that true conceptions represent perceptions in their real order.

There was a superstition once prevalent that if a sorcerer constructed a waxen image of any man, all the operations he performed on that image would be simultaneously effected on the man; so that pricking a pin in the waxen breast was equivalent to planting a dagger in the man's. It is an analogous superstition that operations performed on thoughts are equivalent to operations performed on things, and that we have only to look inwards to see the process that goes on outwards. The analogy may be carried further. The operation performed on the waxen image does represent what would be the result of a similar operation performed on the man, but to what extent? only to the extent in which the image and the man are equivalent, — i. e. wherein both are material forms destroyed by the agents. But in all other respects — in those wherein they differ as waxen substance and living organism—the parallelism fails. Thus the logical operations on conceptions may represent similar operations on perceptions, the interpretation of an ideal construction is a valid interpretation of the external order, in so far, and only in so far, as the one can be taken for the rational equivalent of the other. But this is precisely the domain of Verification.

Starting from the admission that Philosophy is ideal construction formed out of symbols which represent, or are intended to represent, the real order in Feeling, and can only be true when these symbols are the equivalents of their significates, we must reject Hegel's Method, which proceeds on a reversal of this relation between

Thought and Feeling, and declares Thought to be prior in nature, though posterior in time, preceding Feeling as the abstract precedes the concrete. Analysis having once reached the abstract, and seen it everywhere throughout the concretes, Hegel concludes that the abstract was before the concretes, they being simply its concretions, and it not being an abstraction of what is common to them. This, as I have said before, is the fallacy of erecting a result into a principle, making the end the origin.*

We shall have to return to this point presently, but must here continue our survey of the two Methods in their agreements and divergences. Lagrange admits that the principle of virtual velocities is not sufficiently evident in itself to be erected into a first principle, but urges that, nevertheless, it may be regarded as the general expression of the laws of equilibrium. Hegel would also have admitted that his principle of the dialectic process is not selfevident, but would urge that, when reached by analysis of the movement of Thought, it may be recognized as the most general expression of all logical operations, and (since Nature is but the objective aspect of Thought) of all natural processes. So far the two Methods agree. But our next step confronts an important variation. The principle of virtual velocities is seen, when expounded, to be irresistible: it is reducible to an identical proposition. The

* "Were not the dicta of Locke and Hegel, though apparently a reversal the one of the other, after all identical? Locke says, Notions are abstractions from Sensations; while for his part Hegel says, Sensations are concretions from Notions: where at bottom is the difference? Yes, but observe, Hegel's series is the organic system of Thought complete, so to speak, alive in itself." (STIRLING, Secret of Hegel, 1865, I. 163.) Locke's series is quite as organic as Hegel's; and Mr. Stirling has indicated, in a subsequent passage, where the important difference lies, namely, "that Thought never could have been acquired without previous sensuous experience. Yes, but what matters that? We do not wish it to be subjective Thought; it is objective Thought; it is Thought really out there, if you will, in that incrustation that is named the world. It, this world, and all outer objects, are but sensuous congeries, sensuous incrustations of these thoughts. Did a human subject not exist, it is conceivable that this congeries and incrustation would still exist, and it would exist still as a congeries and incrustation of objective Thought." This transporting of Thought out of the organism into the External Order this transfiguration of Existence into a gigantic Ego, a thinking universe, which is man "writ large — is the very fallacy arising from converting resultants into principles.