is a pure intuition, being a singular concept, not made up by sensations, but itself the fundamental form of all external sensation. This pure intuition is in fact easily perceived in geometrical axioms, and any mental construction of postulates or even problems. That in space there are no more than three dimensions, that between two points there is but one straight line, that in a plane surface from a given point with a given right line a circle is describable, are not conclusions from some universal notion of space, but only discernible in space as in the concrete. Which things in a given space lie toward one side and which are turned toward the other can by no acuteness of reasoning be described discursively or reduced to intellectual marks. There being in perfectly similar and equal but incongruous solids, such as the right and the left hand, conceived of solely as to extent, or spherical triangles in opposite hemispheres, a difference rendering impossible the coincidence of their limits of extension, although for all that can be stated in marks intelligible to the mind by speech they are interchangeable, it is patent that only by pure intuition can the difference, namely, incongruity, be noticed. Geometry, therefore, uses principles not only undoubted and discursive but falling under the mental view, and the obviousness of its demonstrations—which means the clearness of certain cognition in as far as assimilated to sensual knowledge—is not only greatest, but the only one which is given in the pure sciences, and the exemplar and medium of all obviousness in the others. For, since geometry considers the relations of space, the concept of which contains the very form of all sensual intuition, nothing that is perceived by the external sense can be clear and perspicuous unless by means of that intuition which it is the business of geometry to contemplate. Besides, this science does not demonstrate its universal propositions by thinking the object through the universal concept, as is done in intellectual disquisition, but by submitting it to the eyes in a single intuition, as is done in matters of sense.[6]
D. Space is not something objective and real, neither substance, nor accident, nor relation; but subjective and ideal, arising by fixed law from the nature of the mind like an outline for the mutual co-ordination of all external sensations whatsoever. Those who defend the reality of space either conceive of it as an absolute and immense receptacle of possible things, an opinion which, besides the English, pleases most geometricians, or they contend for its being the relation of existing things itself, which clearly vanishes in the removal of things and is thinkable only in actual things, as besides Leibnitz, is maintained by most of our countrymen. The first inane fiction of the reason, imagining true infinite relation without any mutually related things, pertains to the world of fable. But the adherents of the second opinion fall into a much worse error. Whilst the former only cast an obstacle in the way of some rational or noumenal concepts, otherwise most recondite, such as questions concerning the spiritual world, omnipresence, etc., the latter place themselves in fiat opposition to the very phenomena, and to the most faithful interpreter of all phenomena, to geometry. For, not to enlarge upon the obvious circle in which they become involved in defining space, they cast forth geometry, thrown down from the pinnacle of certitude, into the number of those sciences whose principles are empirical. If we have obtained all the properties of space by experience from external relations only, geometrical axioms have only comparative universality, such as is acquired by induction. They have universality evident as far as observed, but neither necessity, except as far as the laws of nature may be established, nor precision, except what is arbitrarily made. There is hope, as in empirical sciences, that a space may some time be discovered endowed with other primary properties, perchance even a rectilinear figure of two lines.
E. Though the concept of space as an objective and real thing or quality is imaginary, it is nevertheless in respect to all sensible things not only perfectly true, it is the foundation of truth in external sensibility. Things cannot appear to the senses under any form but by means of a power of the soul co-ordinating all sensations in accordance with a fixed law implanted in its nature. Since, therefore, nothing at all can be given the senses except conformably to the primary axioms of space and their consequences which are taught by geometry, though their principle be but subjective, yet the soul will necessarily agree with them, since to this extent it agrees with itself; and the laws of sensuality will be the laws of nature so far as it can be perceived by our senses. Nature, therefore, is subject with absolute precision to all the precepts of geometry as to all the properties of space there demonstrated, this being the subjective condition, not hypothetically but intuitively given, of every phenomenon in which nature can ever be revealed to the senses. Surely, unless the concept of space were originally given by the nature of the mind, so as to cause him to toil in vain who should labor to fashion mentally any relations other than those prescribed by it, since in the fiction he would be compelled to employ the aid of this very same concept, geometry could not be used very safely in natural philosophy, For it might be doubted whether this same notion drawn from experience would agree sufficiently with nature, the determinations from which it was abstracted being, perchance, denied, a suspicion of which has entered some minds already. Space, then, is the absolutely first formal principle of the sensible world, not only because by its concept the objects of the universe can be phenomena, but especially for the reason that it is. essentially but one, comprising all externally sensible things whatsoever; and hence constitutes the principle of the universe, that is, of that whole which cannot be the part of another.
COROLLARY
Here, then, are two principles of sensuous cognition, not, as in intellectual knowledge, general concepts, but single and nevertheless pure intuition, in which the parts, and especially the simple parts, do not, as the laws of reason prescribe, contain the possibility of the composite, but, according to the pattern of sensuous intuition, the infinite contains the reason of the part, and finally of its thinkable simple part or rather limit. For unless infinite space as well as infinite time be given, no definite space and time is assignable by limitation, and a point as well as a moment is unthinkable by itself and only conceived in a space and time already given as the limits. All primitive properties of these concepts are then beyond the purview of reason, and hence cannot intellectually be explained in any way. Nevertheless, they are what underlies the intellect when from intuitive primary data it derives consequences according to logical laws with the greatest possible certainty. One of these concepts properly concerns the intuition of the object; the other the state, especially the representative state. Hence space is employed as the type even of the concept of time itself, representing it by a line, and its limits—moments—by points. Time, on the other hand, approaches more to a universal and rational concept, comprising under its relations all things whatsoever, to wit, space itself, and besides, those accidents which are not comprehended in the relations of space, such as the thoughts of the soul. Again, time, besides this, though it certainly does not dictate the laws of reason, yet constitutes the principal conditions under favor of which the mind compares its notions according to the laws of reason. Thus, I cannot judge what is impossible except by predicating a and not-a of the same subject at the same time. And especially, considering experience, though the reference of cause to effect in external objects were to lack the relations of space, still in all things, external or internal, the mind could by the auxiliary relation of time alone be informed which is the first and which latter or caused. And even the quantity of space itself cannot be rendered intelligible unless, referring it to measure as to a unity, we set it forth in number, which itself is but multiplicity distinctly cognized by numeration, that is, by the successive addition of one to one in a given time.
Lastly, the question will arise in any one as if spontaneously, whether either concept be connate or acquired. The latter by what has been shown seems refuted already, but the former, smoothing the way for lazy philosophy, declaring vain by the citing of a first cause any further quest, is not to be admitted thus rashly. But beyond doubt either concept is acquired, not, it is true, abstracted from the sense of objects, for sensation gives the matter not the form of human cognition, but from the very action of the mind co-ordinating its sense-percepts in accordance with perpetual laws, as though an immutable type, and hence to be known intuitively. For sensations excite this
