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points we can draw more than three lines perpendicular to one another. By physically real I mean an entity the existence of which is measured by time and determining or occupying a position C such that we can always fix two points A and B in the physical space so that a moving point P cannot pass from A to B without passing through C.

The proof is contained in a relation binding together the two expressions of a physical fact.* This fact is an hyperacceleration, which is a physical fact as much as the velocity itself, of which the hyperacceleration represents the change within an interval of an infinitesimal order higher and higher than that of the velocity. The analytical expression of it† represents this fact decomposed into its elements. These elements are physical facts themselves, as their addition has for result the whole of the fact itself. They are directed quantities, their directions being for the first three elements, three directions perpendicular to one another in the ordinary space, and for the other components from the fourth upwards, each of them has a direction perpendicular to all those which precede it in the order of the sequence. All this is mathematically established in the theorem embodied in the relation (2)* The meaning of the terms of this relation is very plain for the first three terms, they are quantities physically real. The others from the fourth upwards are also physically real, because they are elements of a fact which is so, and which would lose its integrity and become another phenomenon altogether, if the terms from the fourth upwards were not real, and if they were not such that when added together they would not form a fact physically real. The hyperdimensional terms (4th, 5th, ..., nth) are not in ordinary space, still as elements of a physical fact they must be connected with physical space. Such connexion is, as already mentioned, such that we can fix two points, A and B, of the ordinary space,

* See note at the end, formula (2).

+ See note, formula (1).

correspondent respectively to the beginning and the end of the infinitesimal interval (dt)" (n expressing the infinitesimal order: 4, 5, 6, ..., n), so that the point P moving from A to B, with a law admitting hyperaccelerations, according to the conditions of the theorem, must occupy during the said interval the positions determined by the hyperdimensions.

Thus by a mathematical reasoning alone we are led into the presence of physical entities, which, while they possess the characters of physical reality upon the evidence of a logical process, the only evidence available in this case, still are unperceived. Are these entities still real or do they turn into shadows of reality since they cannot be perceived? This problem, which is strictly one in the domain of philosophy, must be met by the mathematician, who, led by the logic of mathematics, is brought in contact with such entities. While hyperdimensions were only mathematical abstractions chosen and made without any reference to physical reality, but only for the sake of giving generality to the methods and theorems of analytical geometry and mechanics of three dimensions, then the mathematician knew the realm of such entities: they did not belong to this world, they existed in a fairy land. But when we meet them in the analysis of a physical phenomenon then, if we want to keep our faith in mathematical analysis, which has been such a powerful instrument of discovery, we must consider their claim in the physical world, even if we are bound to modify our conception of reality in finding a home for them.

II.

From the meaning which I have fixed of something physically real, it is evident that this is defined in terms of time and physical space, taken as the highest terms, without passing the limits or touching the question as to the reality of these terms. I intend to avoid any inquiry concerning the reality of time and space, and of "what is reality" in its widest sense. I assume physical reality in the meaning given

above; to it belong the dimensions of ordinary space, and, according to the theorem expressed by the relation (2), the fact is established that dimensions and hyperdimensions belong to the physical world upon the same evidence, which is the evidence attached to the conclusion of a mathematical reasoning.

The said relation links, so to speak, the three dimensions of the ordinary space to the hyperdimensions in the expression of the same physical fact, and through them the reality of this fact is seen to continue beyond the boundaries of the perceptual space.

The evidence of their physical reality is the same for dimensions and hyperdimensions, as far as this evidence is given by the logical process which forms the proof of the theorem. Besides this, ordinary dimensions possess also the evidence derived from the perception corresponding to them. Obviously the evidence from perception adds itself to that from the mathematical reasoning, without either increasing the strength of the other in establishing the final conclusion aimed at by both of them. They remain independent from one another, each one sufficient to itself, to establish the reality of the fact, I should say, with an equal amount of light, if they are sufficient to expel any doubt of error in the process with which they evolve from their respective sources. It is only where the probability of error affects either of these two processes that they support each other in establishing the final conclusion common to both.

Thus if I see this table in front of me, the evidence that someone has put it here, is derived from such a principle that there is no need of any perception to confirm it. Logical evidence is not the only one which applies to unperceived objects, there is also another kind tending to establish the physical reality of unperceived objects. A perceiving process and a logical process have both in common the same prerogative of being what may be termed a sign of reality. The object or fact pointed by them has evidence of reality only in so far as it is pointed by these signs.

The evidence is good or bad relatively to the sign being good or bad or rather correct or mistaken, and if we could, by some means, reach a criterion of correctness of these processes we could define through them the reality of the object indicated by them. Perception as a sign of reality points to an object which is either actually given in the act of perception or so connected with the latter that the content of actual perception would lose the meaning of the reality which it otherwise possesses, and the act itself of perception would remain disconnected from the conditions by which its constitutive elements are real. Thus if in perception we perceive an object occupying space and lasting in time, with the moment in which its timely existence is given, there is given also with an evidence equally convincing, that the object has existed in some way through a series of previous moments, and it is connected in some way with a space which is not actually perceived. Our conceptions of time and space as assumed in physical reality are such as not to allow any distinction between the value of evidence given as to the existence of an object actually perceived and as to its existence in space and time essentially connected with the space and moment within the act of perception. In other words if we admit something to be real at a certain moment the evidence of that reality goes beyond the limits of that moment, and affects all that which I have defined as physically real. Thus if we consider the space described by a moving point P, taking motion merely as a correspondence between a determined interval of time, however small it may be, and a determined space, locus of the positions of P, during the said interval, if at the moment that motion is perceived, there is a correspondence between a real interval of time and a real tract of space, the reality of these two elements points to a reality at a previous moment of the correspondence, without which the actual moment would be impossible. So that if real space corresponds to a certain interval of time, however small this may be, a real space must

correspond also to any previous or smaller interval of time contained in it. To admit that the properties or characteristics of space which are recognized within an interval at of the motion, are the same as those within an infinitesimal interval of higher order, cannot be assumed as an axiom, and if accepted as a postulate it can be done, only provisionally, while there is no reason against it. It is just against such postulate that the theorem in question directs its consequences. It shows also how far lies the probability that in motion in general there is an instant in which the body is at rest or moves uniformly. To make an assumption of this kind is to fix a characteristic of the motion. This is what Newton did by assuming that the acceleration was constant and that therefore within an interval of time equal, to (dt)3 the body moved uniformly. We cannot make any hypothesis about the motion within an interval of time, however small this may be, while the nature of motion is given by the law which it follows. It is to this analysis of motion within infinitesimal of infinitesimal intervals that the theorem lends itself. Before I give a description of the process followed in the proof I will briefly state the works and results of a few of the most eminent mathematicians, and I will try to approximate their results to mine, in order to see whether and how it constitutes a really new step forward.

III.

Mathematicians have built logical models of space taking their first move from the characteristics of perceptual space and by means of mathematical analysis have reached general concepts of which the three-dimensional space, the one to which perception corresponds, is only a particular case.* The researches on

* An idea of the large number of researches on this subject can be gathered from the valuable monograph of Professor Gino Loria, “Il presente ed il passato delle teorie geometriche." From Gauss's Disquisitiones, are inspired the memoirs of Riemann, "Ueber die Hypothesen welche der Geometrie zu Grunde liegen," and of Beltrami, and for sixty

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