The Universal Spherical Vortex
To return to the complex vertical system considered in the last chapter, it will be seen that the three-dimensional space which is filled by this system is not "isotropic" about the point that is taken as its centre : in other words, owing to the fixing of one particular and so to speak " privileged " direction which is the axis of the system, namely the vertical, the figure is not homogeneous in all directions from that centre. On the other hand, in the horizontal plane, when we were simultaneously considering all positions of the spiral about the centre, this plane was envisaged homogeneously and under an " isotropic" aspect in respect of its centre. For this to hold good in three-dimensional space, it must be noted that every straight line passing through the centre could be taken as the axis of a system such as the one we have been considering, so that any direction can play the part of the vertical direction; similarly since any plane that passes through the centre is perpendicular to one of these straight lines, it follows that, correlatively, any direction can play the part of the horizontal direction, or indeed of the direction parallel to any one of the three planes of coordinates. In fact, any plane that passes through the centre can become one of these three planes in an indefinite multitude of systems of tri-rectangular coordinates, for it contains an indefinitude of pairs of orthogonal straight lines intersecting at the centre (these lines being all the radii issuing from the pole in the depiction of the spiral) ; and each of these pairs can form any two of the three axes of one of these systems. Just as every point in the space is a potential centre, as was said earlier, so any straight line in that space is a potential axis, and, even when the centre has already been fixed, each straight line that
passes through it is still potentially any one of the three axes. When the central or principal axis of a system has been chosen, it still remains to fix the other two axes in the plane perpendicular to the first and likewise passing through the centre; but it is necessary for not only the centre itself but also the three axes to be determined before the cross can be actually traced, that is, before the entire space can be really measured in its three dimensions.
All systems such as our vertical representation can be regarded as coexisting and as having respectively as central axes all the straight lines that pass through the centre, for in fact they do coexist in the potential state, and besides, this is no bar to afterwards choosing three particular axes of coordinates to which the whole space will be referred. Here again, all the systems in question are really only different positions of one and the same system as its axis assumes every possible position about the centre, and the systems interpenetrate for the same reason as before, namely that each of them comprises all the points in the space. One might say that it is the principial point previously mentioned (independent of any determination, and representing the being in itself), which effectuates or realizes this space, hitherto potential only and conceived as a mere possibility of development, by filling its total volume, indefinite to the third power, by the complete expansion of its virtualities in all directions. Moreover, it is in the plenitude of expansion that perfect homogeneity is obtained, just as, conversely, extreme distinction is realizable only in extreme universality [^1]; at the central point of the being, as was said earlier, perfect equilibrium is established between the opposing terms of all contrasts and all antinomies to which outward and particular viewpoints give rise.
When all the systems are considered in this manner as coexisting, the directions of space all play the same part and the radiation from the centre outwards may be regarded as spherical, or rather spheroidal. The total volume, as has been shown, is a spheroid extending indefinitely in all directions,
with a surface that is never closed, any more than the curves previously described. Moreover, the plane spiral, when simultaneously envisaged in all its positions, is nothing but a section of that surface by a plane passing through the centre. It has been stated that the realization of a plane in its integrality was expressed by the calculation of a simple integral; here, as a volume and not a surface is in question, the realization of the space in its integrality would be expressed by the calculation of a double integral [^2]; the two arbitrary constants that would enter into this calculation could be determined by choosing two axes of coordinates, the third axis being thereby fixed, since it must be perpendicular to the plane of the two others and must pass through the centre. It should further be observed that the deployment of this spheroid is ultimately nothing other than the indefinite propagation of a vibratory movement (or "undulatory", for these two terms are ultimately synonymous), no longer in a horizontal plane only, but in the whole three-dimensional space, of which the startingpoint of this movement may now be regarded as the centre. If this space is regarded as a geometrical, that is, spatial symbol of total Possibility (a necessarily imperfect, symbol, because limited by its very nature), then the representation at which we have finally arrived will be the depiction-in so far as such a thing is possible-of the universal spherical vortex by which the realization of all things is accomplished, and which the metaphysical tradition of the Far East calls Tao, that is, the "Way".