rise to questions concerning both their systematic connexion and their origin and meaning. Besides the axioms applicable to quantities of any sort, there are others stating properties of space. The chief difficulty was always felt about the eleventh axiom, which, involving the notion of parallels and the sum of the angles in a triangle, stood altogether apart, and could neither be dispensed with nor logically connected with any other fundamental dictum. At last, from the speculations of Lobatschewsky, Bolyai, and Gauss, it appeared that a consistent geometrical doctrine may be evolved from an hypothesis in contradiction with the said axiom; and soon after, the late Bernhard Riemann discussed space in general and showed that those contradictory systems, Euclidean and Imaginary Geometry, are the expressions of different kinds of space that we may in turn assume as existing. Continued chiefly by Beltrami and Helmholtz, these researches have finally led to a comprehensive theory that may be termed Pangeometry, and is discussed at length in Dr. Erdmann's second chapter (pp. 34-88). Although admitting that our presentation of space is an intuition sui generis, the author observes that we have in particular not only intuitions of lines, triangles, &c., but concepts of such as geometrical species, and that these may be treated not only as concepts of space but of quantity, so as to be determinable by algebraic formulas. Now in the same manner we may form a concept of our universal space as of a quantity: viz., a continuous quantity, of which the elements are universally determined by three commensurable variables, and with a constant measure of curvature equal to zero. Generalising the number of variables into n, and omitting the amount of the curvature, we obtain a summum genus of spaces with a constant curvature, and hence, by replacing the number three, we come down to a concept that embraces our own space as well as spherical space with a positive, and pseudospherical space with a negative, measure of curvature. So Pangeometry branches out into different alternatives, of which Euclidean geometry is but one, and by the light of this discovery the traditional axioms may be reconstructed so as to determine the kind of space considered in our usual geometry, and provide the elements for its production. Restoring the intuitional character which we disregarded while treating of space merely as a sort of quantity, our common space is described as "a threefold extended complexity, congruent in itself and flat (endless)". All this would be perfectly clear, if we were only taught how to conceive a measure of curvature without recurring to intuition.

In a third chapter (pp. 89-135) the philosophical consequences of the new doctrine are discussed. Thinkers now all but universally admit that our spacial presentation cannot be the simple repetition of an arrangement of real things that affect our senses. On the other hand they will be ready to concede that it is dependent not only on a predisposition or capacity of our mind, but also on the nature of impressions that prompt the mind to form its presentations. From the fact of Pangeometry we learn that our mental predisposition in itself admits of more than one sort of spacial intuition, so that it

must needs be the impressions that determine the specifically Euclidean properties of our actual image of the world. So much for empirism as against nativism in psychology. Turning to the epistemological question, Dr. Erdmann shows that our presentations may be viewed either as dependent on or independent of the things represented; and, in another respect, either as faithful likenesses of things, or as reproductions only of their form (say, of quantitative relations of space, time, and law), or else, as mere indications of their presence, that vary together with the nature of things, while differing from them in nature altogether. Accordingly, "empirism" may be subdivided into sensualism, formal empirism, and apriorism; and, on the same principle, "rationalism" into the doctrine of pre-established harmony, formal rationalism, and absolute rationalism or nativism. It is true that few historical systems tally exactly with these distinctions. Most of them offer a compromise between two or more of the typical doctrines, which it will be well, for clearness' sake, to resolve into its constituents. So much we see already, that the modern geometry is incompatible with any kind of "rationalism". Choosing between the possible sorts of " empirism," Riemann and Helmholtz declare for its formal variety, whereas our author offers his reasons for adopting what he terms apriorism, though he grants that even sensualism, "the naïve assumption of the unscientific consciousness," is not repudiated by the new mathematics.

In the fourth chapter (pp. 136-174) Dr. Erdmann draws up his conclusions in the regular form of a philosophical theory of geometry. His readers will find that there are few philosophical treatises, especially in German, so skilfully arranged and neatly worded as this little book, to the merits of which it is impossible to do full justice in a brief notice like the present.* Of course, the author will not expect to have silenced all opponents. As such we may already point out A. Weissenborn in Avenarius's Vierteljahrschrift (II., 2 and 3), and Albrecht Krause in a separate publication. On his own part, the present writer may be excused for briefly stating the cardinal doubts that remain with him unshaken even after both Dr. Erdmann's monograph and Prof. Helmholtz's second paper (in MIND X.).

To borrow the terminology just explained, the characteristic feature of Kant's space-theory appears to be not "rationalism" but "apriorism". In the third section of his last article, Prof. Helm

*It is hardly worth while to enumerate oversights. P. 51, in fine, the words Nenner and Zähler ought to change places; cf. p. 57. P. 90, 1. 10, read der eine jener Fragen. Of material difficulties I mention merely as an example the "rigorous definition" on p. 155: "A straight line is one of which every linear element has zero for its constant measure of curvature". Now, a measure of curvature, according to pp. 51 and 57, is conceived by means of radii, and how to conceive radii except as a kind of straight

lines?

+Kant und Helmholtz über den Ursprung und die Bedeutung der Raumanschauung und der geometrischen Axiome. Lahr: Schauenburg, 1878.

See, e.g., Kant's Werke, III., p. 154, Rosenkr: "All knowledge of

holtz himself shifts his position for a moment from "formal" to "aprioristic empirism". That sensual experience is a conditio sine qua non for the actual occurrence of spacial intuitions, is also a point of Kantian doctrine.* And Dr. Erdmann adopts Prof. Hemholtz's statement, that Kant differs from the true nativists in this respect, that he only avers space-intuition to be a form of our receptivity, without assuming particular spacial intuitions as innate. Up to this point it would seem that we all four agree. Only, the new geometry (of which Kant had a presentiment as early as 1746, Werke, V., p. 27) appears to both our Berlin thinkers to open a prospect towards a more precise distribution of parts between mental constitution and outward influences, which I must persist in considering as wholly delusive. Supposing it could be proved, as they contend, that our mind taken by itself is equally open to the intuitions of all or several of the kinds of space defined by Pangeometry, then indeed it would follow that our actual beholding of a world in Euclidean space ought to depend on something in the impressions that codetermine our consciousness from without. But the difficulty is that the proof we require cannot really be given. First of all, in this order of investigations we have no right to appeal to physiological or psychophysical research, however admirably conducted, because this necessarily proceeds on the assumption of objective space, and the appeal is of no avail unless we could consider objective space as equivalent in some sense to absolutely real space,-which, as involving a begging of the question, we are not at liberty to do. Speaking critically, as we must do in this case, a space inhabited by an intelligent being cannot be shown to have any connexion with a space conceived in that being's mind. So the one legitimate way open to our speculators is to argue from the possibility of imagining other relations of space beside those of Euclid. Against such arguing I need not urge that this possibility is, even in the case of Prof. Helmholtz, but a very limited one. Even if fully admitting that the imaginative powers of highly cultivated men may be expanded so far as to embrace spherical, pseudospherical, and perhaps other spaces of three dimensions with the same case as that to which we are all accustomed, we should be compelled to ask whether they owed that expansion to an emancipation from the narrowing influence of constant Euclidean experience, or rather to a more advanced development from the data of Euclidean experience itself. There is a vast difference between the notion of what our mind may be

things merely from pure intellect or pure reason is nothing but appearance, and truth is only in experience".

*Werke, II., p. 340, Rosenkr: "Space-considered before any things that determine (fill or limit) it, or rather, which give an empirical intuition in accordance with its form--is (under the name of absolute space) nothing but the naked possibility of outward phenomena. . The empirical intuition is not a compound from phenomena and space (or observation and empty intuition), but both are combined in one and the same empirical intuition, as its matter and form respectively."

† Erdmann, p. 105; Helmholtz, Handb. der physiol. Optik, p. 441.

in itself capable to perform, and what it may be trained to achieve subject to the express condition of beginning with an experience that provides an Euclidean basis and no other. Anybody studying those modern mathematical theories-of which I nowise would disparage the technical importance-will perceive their continuous generation out of the old geometry. Indeed it is from Drs. Helmholtz and Erdmann themselves that we learn to understand them in that light. With the facts before us, why should we hold Pangeometry to exist by virtue of our mental nature in spite of habits acquired by impressions from without, rather than to be a logical outgrowth of Euclidean geometry, which latter we acknowledge as the joint produce of mind and impressions? It is true that in the latter case the old question returns, unsolved as before: How much in our experience is due to the nature of mind and to solicitations from the outside respectively? But it may be better for philosophy to recognise this present state of things than impatiently to accept from physical science a sort of solvitur ambulando. J. P. N. LAND.

VII.-REPORTS.

Consciousness under Chloroform.-Under this title Mr. Spencer has just added to the Appendix of Vol. I. of the Principles of Psychology, the following remarkable record of experience with his observations on it :

A University graduate whose studies in Psychology and Philosophy have made him an observer able to see the meanings of his experiences, has furnished me with the following account of the feelings and ideas that arose in him during loss of consciousness and during return to consciousness. My correspondent, describing himself as extremely susceptible to female beauty, explains that "the girl" named in the course of the description was an unknown young lady in a railway carriage which brought him up to town to the dentist's. He says his system resisted the influence of chloroform to such a degree, that it took twenty minutes to produce insensibility: the result being that for a much longer time than usual he underwent partial hyperesthesia instead of anaesthesia. After specifying some dreadI began to ful sensations which soon arose he goes on to say:be terrified to such a wonderful extent as I would never before have guessed possible. I made an involuntary effort to get out of the chair, and thensuddenly became aware that I was looking at nothing: while taken up by the confusion in my lungs, the outward things in the room had gone, and I I felt a force on my arm (which did not strike was alone in the dark". me as the surgeon's 'hand,' but merely as an external restraint) keeping me down, and this was the last straw which made me give in, the last definite thing (smell, sound, sight or touch) I remembered outside my own body. Instantly I was seized and overwhelmed by the panic inside. I could feel every air-cell struggling spasmodically against an awful pressure. In their struggle they seemed to tear away from one another in all directions, and there was universal racking torture, while meantime the common foe, in the shape of this iron pressure, kept settling down with more and more irresistible might into every nook and crevice of the scene. My consciousness was now about this: I was not aware of anything but an isolated

scene of torture, pervaded by a hitherto unknown sense of terror (and by what I have since learnt is called the unity of consciousness': this never deserted the scene, even down to the very last inaudible heart-beat). Yet I call it a 'scene,' because I recognised some different parts of my body, and felt that the pain in one part was not the same as that in another. Meanwhile, along with the increased intensity of convulsion in my lungs, an element of noise had sprung up. A chaotic roaring ran through my brain, innumerable drums began to beat far inside my ear, till the confusion presently came to a monstrous thudding, every thud of which wounded me like a club falling repeatedly on the same spot.

"From this stage my lungs ceased to occupy me, and I forget how the struggle finished. There was a sense of comparative relief that, at any rate, one force was victorious, and the distraction over; the strange large fright that had seized me so entirely when I felt myself ensnared into dark suffocation was now gone also, and there was only left the huge thudding at my ears, and the terribly impetuous stroke of my heart. The thudding gradually got less acutely painful, and less loud; I remember a recognition of satisfaction that one more fearful disturbance was gone. But, while the thunder in my ear was thus growing duller, all of a sudden my heart sprang out with a more vivid flash of sensation than any of those previous ones. The force of an express engine was straining there, and like a burning ball it leapt from side to side, faster and faster, hitting me with such superhuman earnestness that I felt each time as if the iron had entered my soul, and it was all over with me for ever. (Not that 'I' was now any more than this burning hot heart and the walled space in which it was making its strokes : the rest of 'me' had gone unobserved out of focus.) Every stroke produced exquisite pain on the flesh against which it beat glowing, and there was a radiation, as from a molten lump of metal between enclosures. Presently the unbearable heat got less, and there was nothing remaining except a pendulous movement, slackening speed, and not painful. Of nothing beyond was I conscious but this warm body vibrating: not a single other part of me was left, and there was not a single other movement of any sort to attract my attention. A fading sense of infinite leisure at last, in a dreamy inaudible air; then all was hushed out of notice.

66

There was the breaking of a silence that might have been going on for ever in the utterly dark air. An undisturbed empty quiet was everywhere, except that a stupid presence lay like a heavy intrusion somewhere, a blotch on the calm. This blotch became more inharmonious, more distinctly leaden; it was a heavier pressure,—it is actually intruding further, and before almost there was time to wonder feebly how disagreeable was this interruption of untroubled quiet, it had loomed out as something unspeakably cruel and woeful. For a bit there was nothing more than this profoundly cruel presence, and my recognition of it. It seemed unutterably monstrous in its nature, and I felt it like some superhuman injustice; but so entire had been the still rest all round before its shadow troubled me, that I had no notion of making the faintest remonstrance. It got worse.

Just as the cruelty and injustice became so unbearable that I hardly could take it in, suddenly it came out a massive, pulsating pain, and I was all over one tender wound, with this dense pain probing me to my deepest depths. I felt one sympathetic body of atoms, and at each probe of the pain every single atom was forced by a tremendous pressure into all the rest, while everyone of them was acutely tender, and shrank from the wound-only there was nowhere to shrink. A little before, I had merely

*If there were a noun belonging to the verb 'to be aware of' like 'recognition' to 'recognise,' it would be the one to use here.