CHAPTER XV Representation of the Continuity of the different Modalities of One and the same State of the Being
If we consider one of the being's states, depicted by a horizontal plane in the " microcosmic" representation that we have described, it remains to say more precisely what the centre of this plane and also the vertical axis that passes through this centre correspond to. But, to reach that point, it will be necessary to introduce a further geometrical representation, which will show not only, as hitherto, the parallelism or correspondence, but also the continuity which exists between the modalities of each state as well as between the different states themselves.
For this purpose, the figure will have to undergo a change, which corresponds to what in analytical geometry is termed a passage from a system of rectilinear co-ordinates to a system of polar co-ordinates. Instead of representing different modalities of one and the same state by parallel straight lines, as previously, we can represent them by concentric circumferences described in the same horizontal plane, and having their common centre at the centre of the plane itself, that is to say, at its meeting-point with the vertical axis.
In this way, it becomes clear that each modality is finite and limited, because it is depicted by a circumference, which is a closed curve, or at least a line whose ends are known and as it were given. [1] On the other hand this circumference contains an indefinite multitude of
points, [2] representing the indefinitude of secondary modifications that are comprised in the modality considered, whatever it may be. [3] Further, the concentric circumferences must leave no interval between one another, apart from the infinitesimal distance between two immediately adjacent points (we shall return to this question a little later), so that the totality of these circumferences will comprise all the points in the plane, which implies that there is continuity between them. However, to achieve a real continuity, the end of each circumference must coincide with the beginning of the following one (and not that of the same circumference) ; and for this to be possible without the two successive circumferences' being confounded, it is necessary that these circumferences, or rather the curves that we have been regarding as such, shall be in actual fact non-closed curves.
Indeed, we can go further in this direction : it is physically impossible in fact to describe a line that is truly a closed curve. To prove this, it is enough to observe that in the space in which our corporeal modality is situated, everything is ceaselessly in motion (owing to the effect of the spatial and temporal conditions, of which motion is as it were a
resultant) ; so that, if we want to describe a circumference, and start at a given point in space, we shall necessarily find ourselves at a different point when we have completed it, and shall never again pass through the starting-point. Similarly, the curve that symbolizes the course of any evolutive [4] cycle will never have to pass twice through one and the same point, which is tantamount to saying that there cannot be a closed curve (nor a curve containing " multiple points "). This representation shows that there cannot be two identical possibilities in the Universe, which indeed would amount to a limitation of total Possibility-an impossible limitation, because, since it would have to contain Possibility, it could not be contained therein. Thus any limitation of universal Possibility is in the strict and proper sense of the word an impossibility; and for this reason all philosophical systems, which, qua systems, explicitly or implicitly postulate such limitations, stand equally condemned from a metaphysical standpoint. [5] To return to identical or supposedly identical possibilities, it should also be pointed out, for greater exactitude, that two possibilities that were truly identical would not differ in respect of any of their conditions of realization; but if all the conditions are the same, then it is also the same possibility and not two distinct ones, since there is then coincidence in all respects. [6] This reasoning can be strictly applied to all the points in our representation, each of these points depicting a particular modification which realizes a certain given possibility. [7]
The beginning and the end of any one of the circumferences we have to consider, then, are not the same point, but two consecutive points on one and the same radius, and in reality they cannot even be said to belong to the same circumference : one still belongs to the preceding one, of which it is the end, and the other to the following one, of which it is the beginning. The extreme terms of an indefinite series can be regarded as situated outside that series, by the very fact that they establish its continuity with other series; and all this can be applied, in particular, to the birth and death of the corporeal modality of the human individuality. Thus, the two extreme modifications of each modality do not coincide, but there is simply correspondence between them in the state of the being of which those modalities form part, this correspondence being indicated by the situation of the points representing them on one and the same radius from the centre of the plane. Consequently, the same radius will contain the extreme modifications of all the modalities of the state in question, but the modalities should not be regarded, properly speaking, as successive (for they can just as well be simultaneous), but only as logically linked together. The curves that depict these modalities, instead of being circumferences as we had originally supposed, are the successive turns of an indefinite spiral described in the horizontal plane and developing outwards from its centre. This curve continuously broadens out, the radius varying by an infinitesimal quantity, namely the distance between two consecutive points on the radius. The distance may be deemed as small as one likes, in accordance with the actual definition of infinitesimal quantities, namely quantities capable of diminishing indefinitely; but it can never be regarded as nil, for the two consecutive points are not confounded; were it able to become nil, then there would no longer be anything but one and the same point.