We have explained the origin and nature of formal ubication; we have yet to point out its division. Ubication may be considered either objectively or subjectively. Objectively considered, it is nothing else than a point in space marked out by a simple point of matter. We say, by a simple point of matter, because distinct material points in space have distinct ubications. Hence, we cannot approve those philosophers who confound the ubi with the locus, that is, the ubication with the place occupied by a body. It is true that those philosophers held the continuity of matter; but they should have seen all the same that all dimensions involved distinct ubications, and that every term designable in such dimensions has an ubication of its own independent of the ubications of every other designable term; which proves that the locus of a body implies a great number of ubications, and therefore cannot be considered as the synonym of ubi.
If the ubication is considered subjectively, that is, as an appurtenance of the subject of which it is predicated, it may be defined as the mode of being of a simple element in space. This mode consists of a mere relativity; for it results from the extrinsic termination of absolute space, as already explained. Hence, the ubication is not received in the subject of which it is predicated, and does not inhere in it, but, like all other relativities and connotations, simply connects it with its correlative, and lies, so to say, between the two.[149]
But, although it consists of a mere relativity, the ubication still admits of being divided into absolute and relative, according as it is conceived absolutely as it is in itself, or compared with other ubications. Nor is this strange; for relative entities can be considered both as to what they are in themselves, and as to what they are to one another. Likeness, for instance, is a relation; and yet when we know the likeness of Peter to Paul, and the likeness of Peter to John, we can still compare the one likeness with the other, and pronounce that the one is greater than the other.
When the ubication is considered simply as a termination of absolute space without regard for anything else, then we call it absolute, and we define it as the mode of being of an element in absolute space, by which the element is constituted in the divine presence. This absolute ubication is an essential mode of the material element no less than its dependence from the first cause, and is altogether immutable so long as the element exists; for, on the one hand, the element cannot exist but within the domain of divine immensity, and, on the other, it cannot have different modes of being with regard to it, as absolute space is the same all throughout, and the element, however much we may try to imagine different positions for it, must always be in the centre, so to say, of that infinite expanse. Hence, absolute ubication is altogether unchangeable.
When the ubication of one element is compared with that of another element in order to ascertain their mutual relation in space, then the ubication is called relative, and, as such, it may be defined as the mode of terminating a relation in space. This ubication is changeable, not in its intrinsic entity, but in its relative formality; and it is only under this formality that the ubication can be ranked among the predicamental accidents; for this changeable formality is the only thing in it which bears the stamp of an accidental entity.
The consideration of relative ubications leads us directly to the consideration of the relation existing between two points distinctly ubicated in space. Such a relation is called distance. Distance is commonly considered as a quantity; yet it is not primarily a quantity, but simply the relation existing between two ubications with room for movement from the one to the other. Nevertheless, this very possibility of movement from one point to another gives us a sufficient foundation for considering the relation of distance as a virtual dimensive quantity. For the movement which is possible between two distant points may be greater or less, according to the different manners in which these points are related. Now, more and less imply quantity.
The quantity of distance is essentially continuous. For it is by continuous movement that the length of the distance is measured. The point which by its movement measures the distance, describes a straight line by the shifting of its ubication from one term of the distance to the other. The distance, as a relation, is the object of the intellect, but, as a virtual quantity, it is the object of imagination also. We cannot conceive distances as relations without at the same time apprehending them as quantities. For, as we cannot estimate distances except by the extent of the movement required in order to pass from one of its terms to the other, we always conceive distances as relative quantities of length; and yet distances, objectively, are only relations, by which such quantities of length are determined. The true quantity of length is the line which is drawn, or can be drawn, by the movement of a point from term to term. In fact, a line which reaches from term to term exhibits in itself the extent of the movement by which it is generated, and it may rightly be looked upon as a track of it, inasmuch as the point, which describes it, formally marks by its gliding ubication all the intermediate space. The marking is, of course, a transient act; but transient though it is, it gives to the intermediate space a permanent connotation; for a fact once passed, remains a fact for ever. Thus the gliding ubication leaves a permanent, intelligible, though invisible, mark of its passage; and this we call a geometric line. The line is therefore, formally, a quantity of length, whereas the distance is only virtually a quantity, inasmuch as it determines the length of the movement by which the line can be described. Nevertheless, since we cannot, as already remarked, conceive distances without referring the one of its terms to the other through space, and, therefore, without drawing, at least mentally, a line from the one to the other, all distances, as known to us, are already measured in some manner, and consequently they exhibit themselves as formal quantities. Distance is the base of all dimensions in space, and its extension is measured by movement. It is therefore manifest that no extension in space is conceivable without movement, and all quantity of extension is measured by movement.
We have said that distance is a relation between two terms as existing in distinct ubications; and we have now to inquire what is the foundation of such a relation. This question is of high philosophical importance, as on its solution depends whether some of our arguments against Pantheism are or are not conclusive. Common people, and a great number of philosophers too, confound relations with their foundation, and do not reflect that when they talk of distances as relative spaces, they do not speak with sufficient distinctness.
We are going to show that relative space must be distinguished from distances, as well as from geometric surfaces and volumes, although these quantities are also called “relative spaces” by an improper application of words. Relative space is not an intrinsic constituent, but only an extrinsic foundation, of these relative quantities; hence these quantities cannot be styled “relative spaces” without attributing to the formal results what strictly belongs to their formal reason.
What is relative space? Whoever understands the meaning of the words will say that relative space is that through which the movement from a point to another point is possible. Now, the possibility of movement can be viewed under three different aspects. First, as a possibility dependent on the active power of a mover; for movement is impossible without a mover. Secondly, as a possibility dependent on the passivity of the movable term; for no movement can be imparted to a term which does not receive the momentum. Thirdly, as a possibility dependent on the perviousness of space which allows a free passage to the moving point; for this is absolutely necessary for the possibility of movement.