CHAPTER XVIII Passage from Rectilinear Coordinates to Polar Coordinates: Continuiry by Rofation
It is now necessary to return to the last of the geometrical representations that have been mentioned. The introduction of this is tantamount to substituting polar coordinates for the rectilinear and rectangular coordinates of the previous " microcosmic " representation. Every variation in the radius of the spiral (the latter starting from the centre tangentially to the tion on the axis that traverses all the modalities, that is, the axis perpendicular to the direction in which the development of each modality takes place. As for the variations on the axis parallel to this last direction, these are replaced by the different positions occupied by the radius in revolving about the pole (the centre of the plane or origin of the coordinates), in other words by the variations in its angle of rotation, measured from a given position taken as origin. This initial position, which will be the normal one at the outset of the spiral (the latter starting from the centre tangentially to the radius perpendicular to that position) will be that of the radius which, as already said, contains all the extreme modifications (beginning and end) of all the modalities.
But, of all such modalities, not only do the beginning and the end correspond to each other, but each intermediate modification or element of a modality has likewise its corresponding element in every other, the corresponding modifications being always represented by points lying on one and the same radius issuing from the pole. If this radius, whichever it may be, is taken as normal at the origin of the spiral, we shall always get the same spiral, but the figure as a whole will have turned through a certain angle. In order to represent the
perfect continuity between all the modalities and the correspondence of all their elements, the figure would have to be imagined as simultaneously occupying all possible positions around the pole, with all these figures interpenetrating one another, since each of them, in the sum total of its indefinite development, equally comprises all the points in the plane. Properly speaking, it is only one and the same figure in an indefinitude of different positions, which correspond to the indefinitude of values the angle of rotation can assume, supposing this angle to vary continuously until the radius, starting from the given initial position, returns after a complete revolution to superimpose itself upon that first position.
On that supposition, we should get the exact image of a vibratory movement propagating itself indefinitely, in concentric waves, around its starting-point, in a horizontal plane such as the free surface of a liquid [^1]; and that would be the most exact possible geometrical symbol of the integrality of a state of being. Were it desired to go further into considerations of a purely mathematical order-which are not to the point here except in so far as they furnish symbolical repre-sentations-it could even be shown that the realization of that integrality would correspond to the integration of the differential equation expressing the relationship between the concomitant variations of the radius and of its angle of rotation, both varying together, and one as a function of the other, continuously, that is, by infinitesimal quantities. The arbitrary constant that figures in the integral would be determined by the position of the radius taken as origin, and this same quantity, which is fixed for a given position of the figure, would be bound to vary continuously from 0 to 2π for all its positions; accordingly, if we regard the positions as able to be simultaneous (this amounts to suppressing the temporal condition, which endows the activity of manifestation with the particular qualification constituting movement), the constant must be left indeterminate between those two extreme values.
However, it should be carefully noted that these geometrical representations are always to some extent imperfect, as indeed must be the case with any representation or formal expression. In practice, we are naturally obliged to situate them in a particular space, in a given extension, and space, even when envisaged in the whole extension it is capable of, is no more than a special condition which is contained in one of the degrees of universal Existence, and to which (added to or combined with other conditions of the same order) certain of the multiple domains comprised in that degree of Existence are subjected-each of such domains constituting, in the " macrocosm", the analogue of what in the " microcosm" is the corresponding state of the being, situated at that same degree. The representation is necessarily imperfect, simply by being enclosed within narrower limits than that which it represents, and indeed it would otherwise be useless [^2]. On the other hand, while always remaining included within the. bounds of the at present conceivable; or even the far more restricted bounds of the imaginable (which proceeds wholly from the sensible), the representation will be proportionately less imperfect the less limited it becomes, which really amounts to saying, the higher the power of the indefinite it introduces [^3]. In spatial representations, in particular, this is expressed by adding an extra dimension, as has been shown above; however, this question will be further clarified later.