REPRESENTATION OF THE EQUILIBRIUM OF FORCES
In connection with negative numbers, we shall now speak of the rather disputable consequences of the use of these numbers from the point of view of mechanics, even though this is only a digression with respect to the principal subject of our study; moreover, since in virtue of its object the field of mechanics itself is in reality a physical science, the very fact that it is treated as an integral part of mathematics in consequence of the exclusively quantitative point of view of science today means that some rather singular distortions have been introduced. Let us only say that the so-called 'principles' upon which modern mathematicians would build this science, such as they conceive of it, can be referred to as 'principles' only in a completely abusive manner, as they are in fact only more or less well-founded hypotheses, or again, in the most favorable case, only simple laws that are general to some degree, perhaps more general than others, if one likes, but still having nothing in common with true universal principles; in a science constituted according to the traditional point of view, the laws of mechanics would at most be mere applications of these principles to an even more specialized domain. Without entering into excessively lengthy explanations, let us cite as an example of the first case, the so-called 'principle of inertia', which nothing can justify, neither experience, which on the contrary shows that inertia has no role in nature, nor in the understanding, which cannot conceive of this so-called inertia that consists only in a complete absence of properties; one could only legitimately apply such a word to the pure potentiality of universal
substance, or to the materia prima of the Scholastics, which is moreover for this very reason properly 'unintelligible'; but this materia prima is assuredly something completely different from the 'matter' of the physicists.[1] An example of the second case may be seen in what is called the 'principle of the equality of action and reaction', which is so little a principle that it is immediately deduced from the general law of the equilibrium of natural forces: whenever this equilibrium is disturbed in any way, it immediately tends to re-establish itself, whence a reaction of which the intensity is equivalent to that of the action that provoked it. It is therefore only a simple, particular case of what the Far-Eastern tradition calls 'concordant actions and reactions', a principle that does not concern the corporeal world alone, as do the laws of mechanics, but indeed the totality of manifestation in all its modes and states; and for a moment we propose to dwell precisely on this question of equilibrium and its mathematical representation, for it is important enough in itself to merit a momentary pause.
Two forces in equilibrium are usually represented by two opposed 'vectors', that is, by two line segments of equal length, but aimed in opposite directions: if two forces applied to the same point have the same intensity and fall along the same line, but in opposite directions, they are in equilibrium; as they are then without action at their point of application, it is even commonly said that they cancel each other out, although this ignores the fact that if one of the forces is suppressed, the other will immediately act, which proves that they were never really cancelled in the first place. The forces are characterized by numerical coefficients proportional to their respective intensities, and two forces of opposing direction are given coefficients with different signs, the one positive, the other negative, so that if the one is f, the other will be -f'. In the case we have just considered, in which the two forces are of the same intensity, the coefficients characterizing them must be equal with respect to their 'absolute values'; one then has f=f', from which one can infer as a condition of their equilibrium that f-f'=0, which is to say that the algebraic sum of the two forces, or of the two 'vectors' representing
them, is null, such that equilibrium is thus defined by zero. Zero having been incorrectly regarded by mathematicians as a sort of symbol for nothingness, as we have already said above-as if nothingness could really be symbolized by anything whatsoever-the result seems to be that equilibrium is the state of non-existence, which is a rather strange consequence; it is nevertheless almost certainly for this reason that, instead of saying that two forces in equilibrium neutralize one another, which would be more exact, it is said that they cancel one another, which is contrary to the reality of things, as we have just made clear by a most elementary observation.
The true notion of equilibrium is something else altogether. In order to understand it, it suffices to point out that all natural forces, and not only mechanical forces (which, let us say again, are no more than a very particular case) but forces of the subtle order as well as those of the corporeal order, are either attractive or repulsive; the first can be considered as compressive forces, or forces of contraction, and the second as expansive forces, or forces of dilation,[2] and basically this is no more than an expression in a particular domain of the fundamental cosmic duality itself. It is easy to understand how, given an initially homogenous medium, for every point of compression there will necessarily correspond an expansion at another point, and inversely, such that two centers of force must be envisaged correlatively, each of which could not exist without the other; this is what one can call the law of polarity, which is, in all its various forms, applicable to all natural phenomena, since it, too, derives from the duality of the very principles that preside over all of manifestation; in the specialized domain with which physicists occupy themselves, this law is above all evident in
electrical and magnetic phenomena, but it is by no means limited to them. Now if two forces, the one compressive, the other expansive, act upon the same point, then the condition requisite for them to be in equilibrium, or to neutralize one another, the condition, that is, which when fulfilled will produce neither contraction nor dilation, is that the intensities of the two forces be equivalent; we do not say equal, since the forces are of different species, and since this is moreover a question of a truly qualitative, and not simply quantitative, difference. The forces can be characterized by coefficients proportional to the contraction or dilation they produce, in such a way that if one considers a compressive force and an expansive force together, the first will have a coefficient n>1, and the second a coefficient n'<1; each of these coefficients will be the ratio of the density of the space surrounding the point in consideration, under the action of the corresponding force, to the original density of the same space, which in this regard is taken to be homogenous when not subject to any forces in virtue of a simple application of the principle of sufficient reason.[3] When neither compression nor dilation is produced, the ratio is necessarily equal to one, since the density of the space is unchanged; in order for two forces acting upon a point to be in equilibrium, their resultant must have a coefficient of one. It is easy to see that the coefficient of this resultant is the product, and not, as in the ordinary conception, the sum of the coefficients of the two forces under consideration; these two coefficients, n and n', must therefore each be the inverse of the other: n'=1/n, and we will then have (n)(n')=1 as the condition for equilibrium; equilibrium will thus no longer be defined by zero, but by the unit.[4]
It will be seen that the definition of equilibrium with respect to the unit-its only real definition-corresponds to the fact that the unit occupies the mid-point in the doubly indefinite sequence of
whole numbers and their inverses, while this central position is as it were usurped by zero in the artificial sequence of positive and negative numbers. Far from being the state of non-existence, equilibrium is on the contrary existence considered in and of itself, independent of its secondary, multiple manifestations; moreover, it is certainly not Non-Being, in the metaphysical sense of the word, for existence, even in this primordial and undifferentiated state, is still the point of departure for all differentiated manifestations, just as the unit is the point of departure for the multiplicity of numbers. As we have just considered it, this unit in which equilibrium resides is what the Far-Eastern tradition calls the 'Invariable Middle'; and according to the same tradition, this equilibrium or harmony is the reflection of the 'Activity of Heaven' at the center of each state, and of each modality of being.