The Irreducible Magnitudes of the Universe

The New Principles: Book II: The Irreducible Magnitudes of the Universe

Chapter I – Time, Space, Matter and Force

1. The Conception of the Irreducible Magnitudes of the Universe

Time, space, matter and force form the elements of things, and the fundamental basis of all our knowledge.

Time and space are the two magnitudes in which we confine the universe. Force is the cause of phenomena, matter their web.

Three of these elements — time, space, and force — are quite irreducible. Matter may be reconverted into force, not only because it is, as I have proved, a particular form of energy, but also because it is only defined, in equations of mechanics, by the symbols fo force (1).

[(1) In the CGS system now generally adopted for the evaluation of the magnitudes of physical quantities, we take into consideration: (1) the fundamental quality, length, mass, and time; and (2) the derived quantities. These last, which are very numerous, comprise notably the derived quantities of geometry — surface, volume, and angle; those of mechanics — speed, acceleration, force, energy, work, power, etc.; and those of electricity and magnetism — resistance, intensity, potential difference, etc.]

Time, space, and force being irreducible, cannot be compared with anything and are indefinable. We only know of them that which our common sense tells us. So soon as, in order to define these great entities, we endeavor to go beyond what is revealed by ordinary observation, we meet with inextricable difficulties and end by acknowledging, as do the philosophers, that they are simply creations of the mind, and cover completely unknown realities.

These realities are not knowable to us, because our senses ever remain interposed between them and us. What we perceive of the universe are only the impressions produced on our senses. The form we give to things is conditioned by the nature of our intelligence. Time and space are, then, subjective notions imposed by our senses on the representation of things, and this is why Kant considered time and space as forms of sensibility. To a superior intelligence, capable of grasping at the same time the order of succession and that of the co-existence of phenomena, our notions of space and time would have no meaning.

It is, moreover, not space and time only, but all phenomena, from matter which we think we know up to the divinities created by our dreams, which have to be considered as forms necessary for our understanding. The world constructed with the impressions of our senses is a summary translation, and necessarily a far from faithful one of the real world which we know not. Time is, for man, nothing but a relation between events. He measures it by the changes in position of a mobile body, such as a star or a clock. It is only by a change, that is to say, by movement, that the notion of time is accessible to us. “In a world void of all kind of movement”, says Kant, “there would not be seen the slightest sequence in the internal state of substances. Hence, the abolition of the relation of substances to one another carries with it the annihilation of sequence and of time”. If there are no events there is evidently no sequence, and consequently no time.

To immobilize the world and the beings which inhabit it would be to immobilize time — that is to say, to cause it to vanish. If this fixedness were absolute, life would be impossible, since life implies change; but neither could anything grow old. The immortal gods who, according to the legends, never undergo change, cannot know time. For them the clock of heaven marks always the same hour. Change is therefore the true generator of time [** Ed.: — see Kozyrev]. It is only conceivable, like forces and all phenomena, under the form of movement. This fundamental concept of movement will be found at the base of all phenomena. It serves to define the magnitudes of the universe, and can only be defined by them. It is not an irreducible concept, for it is formed by the combination of the notions of force, of matter, of space, and of time. It is evident that we require the intervention of all these in order to define the displacement of a body.

In physics the most variations of quantities are expressed by reference to the variations of time. When the curve expressing the relations of a phenomenon with time is known, science can go back from the present to the past and can know the future.

The notion of space is as little clear as that of time. Leibnitz defined it as the order of co-existence of phenomena, time being the order of their succession. Space and time are perhaps two forms of the same thing.

Space does not appear conceivable without the existence of bodies. A world entirely void could not give birth to the idea of space, and this is the reason philosophers refuse to space an objective reality. In their view, space being, like time, a quality, where there is neither phenomenon nor substance, there is neither space nor time.

The above brief expose’ suffices to show how inexact and limited are the ideas man can form as to the fundamental elements of the universe. Our knowledge being only relative, we only define with a known one. All knowledge therefore implies a comparison, but to what can we compare the irreducible elements of things? They condition phenomena, and remain hidden behind them.

If the irreducible magnitudes of the universe are not known in their essence, they at least produce measurable effects. We are situated with regard to them like the railway porter who can weigh with exactness parcels the content of which he is ignorant.

It is of these measurements alone that science is composed. By means of them are established the numerical relations which form to one web of our knowledge, since the realities which uphold them escape us. The properties of things are only properly definable by measurement. The qualitative represents a subjective appreciation which may vary from one individual to another. The quantitative represents a fixed magnitude which can be preserved, and which gives precision to our sensations. The substitution of the quantitative for the qualitative is the principle task of the scholar. “I often say”, writes Lord Kelvin, “that if you measure that of which you speak, and can express it by a number, you know something of your subject; but if you cannot measure it, your knowledge is meager and unsatisfactory”.

2. The Measurement of the Irreducible Magnitudes of the Universe

By measuring and placing one on the other the heterogeneous elements which form the web of things, science has managed to create certain concepts, such as those of mass, kinetic energy, etc., which we have to consider realities by reason of our incapacity to imagine others.

These concepts vary with the way in which we bring together the irreducible elements of things. Associate force with space, and we create the science of energy. Associate space and time, and we create the science of velocities — that is to say, kinematics. Associate force, space and time, and we create the science of mechanical power. It is evident that, by thus acting, we must associate very heterogeneous elements.

Force ( F = MA ) is a coefficient of resistance multiplied by an acceleration. Work ( T = F x E ) is a force multiplied by a length. Velocity ( V = L/T ) is a space divided by a time. Mass (M = P/g) is a weight divided by a velocity, etc.) It is only by the combination of these very different magnitudes, that it has been possible to state precisely the concepts of mechanics on which the interpretation of the phenomena of the universe is still based.

To define completely a phenomenon there have to be associated the three great coordinates of things — time, space, and force. If one or two of these only are measured, the phenomenon is only partially known. The formation of the modern notions of energy and of power furnishes excellent examples of this. They were no precisely stated until to the vague idea of force considered as the synonym of effort was added the notion of space, and then that of time.

In mechanics, force is defined as a cause of movement; the unit of force is represented by the acceleration produced on the unit of mass. When a force displaces its point of application it generates work. This last is the product of the force considered as a cause of movement. The kilogram-meter has been chosen as the unit of work. It is the work necessary to displace a kilogram for the length of a meter. This unit of mechanical energy is now used to measure all forms of energy.

Thus, by the sole fact that we have associated space with force, we can measure this last and comprise it in a formula. This enables us to understand how with an invariable quantity of energy we can produce forces of variable magnitude. If, in fact, we call the Force F, the Space E, and the work T, we obtain according to the preceding definitions T = F x E. In this formula, which defines the unit of work, the force F and the space E can evidently be inversely varied without changing their product — that is to say, the work. We can therefore largely increase the force on condition that we proportionately reduce the space covered. It is this operation which is affected by certain machines, such as the lever, which multiplies the force but not the work. By the expenditure of one kilogram-meter, hundreds of kilogram-meters can be raised, but what is gained in force will be lost in the space covered, and the product F x E will never exceed a kilogram-meter. Force therefore can be multiplied, but not energy, of which the magnitude remains invariable.

Into the unit of work there enter only the elements force and space, but not the element time. One kilogram-meter may be expended in one second or in a thousand years, and the results will necessarily be very different I the two cases. This is very well illustrated in the case of radium, of which one gram contains thousands of kilogram-meters. Such a force appears immense, but its production is in each instant so slight that it would require thousands of years to liberate it entirely. It is the case of a reservoir containing an immense quantity of water which can escape by a drop at a time. Hence, by confining ourselves to the association force and space, we have already created a unit which permits us to evaluate in kilogram-meters the power of any machine moved by any motor, but it does not tell us if these kilogram-meters are produced in one minute or in a year. We know therefore very little of the power of the machine.

To ascertain this, it suffices to superimpose on the two elements force and space, which give us the unit of work, the element time. We shall then have what is called the unit of power, which is the quotient of the work by the time. It shows us the work produced in a given time. If we are told that a machine produces a kilogram-meter, we know nothing as to its power. If it be added that this kilogram-meter is produced in one second, we are fully informed.

The kilogram-meter per second being too small a unit from the commercial point of view, one 75 times larger has been adopted. This is the horsepower, which represents 75 kilograms raised one meter in one second (1).

[(1) In physics other units are often made use of, but we do not alter what has been said. If, instead of being evaluated in kilograms, the force is evaluated in dynes, and if the space, instead of being evaluated in meters, is measured in centimeters, the work, instead of being expressed in kilogram-meters, is expressed in ergs.]

In this last unit are found collected, as will be seen, the three irreducible elements of things — time, space, and force. Matter likewise figures init indirectly, for that which is measured is the force employed to combat its inertia and to give it certain movements.

We have just seen how, by enclosing in space and time that mysterious Proteus called force, it is possible to grasp it and know it under its deceiving forms. On penetrating further into the inmost nature of phenomena, we shall see that space and time not only serve to measure force, but that they also condition its form and its magnitude.

Chapter II – The Great Constants of the Universe: Resistance and Movement

1. Inertia or Resistance to Change

Forces are known to us solely by the movements they generate. Mechanics, which claims to be the foundation of the other sciences and to explain the universe, is devoted to the study of these movements.

The notion of movement implies that of things to move. Observation sows that these things to move present a certain resistance. The resistance of matter to movement or to a change of movement is what is termed its inertia. It is from this property that is derived the notion of mass.

We thus find ourselves in the presence of two elements, not irreducible like those just studied, but fundamental. These are movement and resistance to movement, or, in other words, change and resistance to change. Inertia — that is to say, the aptitude of matter to resist movement or a change in movement — is the most important of its properties, and even the only one which allows us to follow it through all its modifications. While its other characteristics, solidity, color, etc., depend on several variable causes and consequently may change, inertia depends on no factor and is unchangeable. Whether it be liquid, solid or gaseous, whether it be isolated or in combination, the same body possesses an unvarying quantity of inertia. Measured indirectly by the balance, this allows us to follow it through all its changes.

On this notion of the invariability of inertia, or, in other words, of the mass, are based the edifices of chemistry and mechanics. The preponderant part played by inertia in phenomena is a matter of daily observation. It is by virtue of inertia that the worlds continue to circulate in space, that a ball hurled from a cannon by the explosion of gunpowder travels several thousand meters. Inertia being opposed to a change of movement, bodies would even continue their course indefinitely if different antagonistic forces, such as the resistance of the air, did not finally arrest them. A railway train would thus continue to advance with the same velocity without the help of any motor if its inertia did not unceasingly tend to be annulled by various resistances, friction, etc., which the locomotive only serves to overcome. The same inertia of matter forbids the train stopping abruptly. To effect this, very powerful brakes must be employed even if the engine has ceased working. Inertia being opposed to movement as well as to change of movement, it requires a very great force to start the train from its repose, and one equally great to stop it when once in motion.

It results therefore from the principle of inertia that, when a moving body tends to slacken speed from any cause whatsoever, inertia tends to maintain that speed, since, by its definition, it is opposed to change of movement. Conversely, when the speed of the moving body increases, inertia comes in to retard this acceleration for the same reason, viz., that it is opposed to change of movement.

Electricity, which possesses, or at least appears to possess, inertia, behaves like matter in motion. Its inertia acts in the phenomena of induction exactly, as has been said above, by opposing itself to change of movement — that is to say, in the converse direction to the cause which tends to produce its slackening or acceleration. This is expressed by the law of Lenz, which governs the phenomena of induction. It would perhaps be possible to explain them on the principle of the equality of action and reaction without invoking inertia at all. To measure the inertia of matter is easy, to note its properties is likewise easy, but to explain its nature is as yet impossible.

Newton, who was the first to study inertia scientifically, considered it to be a force. “The force which dwells in matter”, he says, “is its power of resistance, and it is by this force that every body perseveres of itself in its actual state of repose or of movement in a straight line”.

At the present day, the tendency is to admit that matter is connected with the ether by lines of force, and that the whole of the inertia of matter should be that of the ether gripped by lines of force. But whether inertia be attributed to matter or to the medium in which it is plunged, this does not bring us any nearer to an explanation.

Perhaps the least improbable thing that may be said regarding inertia is that matter, being, as I have shown, an immense aggregate of forces, possesses certain relations of equilibrium with the ether surrounding it. The movement of a body must break up this equilibrium and create others, from which would result the continuation of the movement and its resistance to change of speed. In the internal equilibrium of a body in motion something is probably changed.

To the notion of inertia there should, doubtless, be attached the principle of the equality of action and reaction. Although this is a fundamental principle in mechanics, it, too, is very little explicable. It has been formulated by Newton as follows: —

“A body exercising on another a pressure or traction, receives from the latter an equal and opposite traction or pressure”. This would signify that if you exercise a traction of 100 kilograms on an infinitely rigid wall it will exercise the same traction on you. The wall thus becomes, as M. Wickersheimer points out, a metaphysical person entering into antagonism with you. At bottom, mechanics, which seems to be the most precise of sciences, the one most foreign to metaphysics, is the one which contains most evident or hidden metaphysical notions. They evidently cover profound but entirely unknown causes. Perhaps we should explain the principle of equal reaction in the direction contrary to action by considering certain forces as couples — that is to say, as acting like a spring stretched between two points. It is evidently impossible then to act on one without the other reacting immediately. Gravity and electricity would come under this head.

2. Mass

The mass which serves to characterize matter is only the measure of its inertia — that is to say, of its resistance to movement. It is measured by seeking the magnitude of the force which must be opposed to inertia in order to annul it. Gravity has been chosen because it is easy to handle. We can by means of weights, each of which represents a certain quantity of attraction, measure the inertia of a certain portion of matter placed on one of the scales of a balance.

The notion of mass was slow in establishing itself. Mach, in his History of Mechanics, points out that Descartes, Newton and Leibnitz had only a very vague comprehension of it. Galileo confused mass with weight, which many people do even at the present time, although by reason of the units adopted, weight is represented by a figure about 10 times greater than that expressed by mass (1).

[(1) The distinction between weight and mass, formerly considered synonymous, only became manifest when the observation of the pendulum revealed that the same body may receive a different acceleration of gravity in different parts of the globe. It was in 1871 that it was noted for the first time in astronomical observations that a clock giving the exact time in Paris no longer did so in Guiana. To render its pace regular, it is necessary to shorten the length of the regulating pendulum.]

The term mass is, moreover, employed at the present day in two different senses. For physicists mass is a coefficient of inertia, and for astronomers a coefficient of attraction. If the attraction due to gravity were the same all over the globe, the mass of a body, that is to say, the quantity of inertia it possesses — would be measured according to the force of attraction necessary to annul it. Chemists, who have only to compare the masses of bodies, proceed in no other way. For the calculations of mechanics it was necessary to find another element, because gravity alters with latitude and the height from the earth. This last variation even shows itself at the different stories of a house.

The weight of a body varies from one place to another, but the acceleration which this body may take undergoes the same variation. The ratio of these two magnitudes is therefore constant at all points of the globe. It is this relation P/g which always figures in the calculations of mechanics. Given the value of the number g, it follows that in numerical expressions the mass of a body hardly represents the tenth part of its weight. The equation M = P/g which defines mass, refers to the gravity; but as the weight may be replaced by any force F, which produces an acceleration A, we obtain as a general expression of mass M = F/A. This is the fundamental equation of mechanics. One must not look too closely into its meaning.

Mass has been considered as an invariable magnitude down to the recent researches mentioned in my last book. These last have shown that not only does the mass vary by the dissociation of atoms, but, further, that the products of this dissociation have a mass varying with their velocity. This mass can even increase to the point of becoming infinite — that is to say, when the velocity approached that of light. Nothing proves, moreover, that it would not be the same with ordinary matter animated by a like velocity.

Not only does the mass vary with the velocity, but it has lately become a question whether it does not also vary with the temperature. The question has not yet been elucidated. However that may be, mass is not at all that invariable magnitude which chemistry and mechanics formerly supposed it to be. The element which science considered as the immovable pivot of phenomena, the starting point to which it endeavored to refer all things, has become a variable magnitude of which the apparent fixity was only due to the imperfection of our means of observation.

The inertia of matter is still, however, the most stable thing in the changing ocean of phenomena. This stability is not absolute, but as regards our ordinary requirements the inertia of matter can be considered as one of the great constants of the universe.

3. Movement and Force

For half a century science thought she had discovered a second constant element in the universe. This element is energy, of which forces would be simple manifestations.

We will now examine only the fundamental elements of forces. They are knowable to us by the movements they produce, and that is why, in the classic mechanics, force is simply defined as a cause of movement.

By virtue of their inertia alone, bodies would only assume a uniform and rectilinear movement. Directly this movement is accelerated, we recognize that a force has intervened. It is solely this acceleration which mechanics measures and which figures in its equations.

Force is therefore only known to mechanics through movement. Movement is not an irreducible magnitude, since it is derived from the four great elements of the universe — time, space, matter and force — which alone enable it to be defined.

We have seen previously how by associating force and space the unit of mechanical energy and of work has been constituted; we shall see in a later chapter the transformations which the modern notion of the conservation of energy has introduced into the conceptions of force.

What precedes shows us how notions of movement and of resistance are derived from those of force and mass, on which the principles of mechanics were built up. The equation F = MA defines force by the acceleration imparted to a body endowed with resistance to movement.

To sum up, movement — that is to say, change — and inertia — that is to say, resistance to change — constitute the fundamental elements accessible to mechanics. We will now see how, by associating them, this science has sought to interpret the phenomena of the universe.

Chapter III – The Building Up of Forces and the Mechanical Explanations of the Universe

1. The Cycle of Forces

We have just seen that on reducing to their essential elements the forces of the universe there still remain resistance and movement. Resistance is represented by the inertia of matter or of the ether, and movement by the displacement of these substances in space and time.

The magnitude of forces is determined by the velocity of movements that they produce, their form by the nature of these movements. The movements of matter are only apparent to us when it comes into contact with an antagonistic factor which annuls or diminishes its velocity. The earth, for instance, by reason of its movements of rotation and of translation in space, possesses an immense kinetic energy; but it is not noticed, because out globe meets no obstacles in its path. Yet its kinetic energy would be sufficient, perhaps, to reduce to vapor any planet it chanced to strike. All things living on the surface of our globe are carried along with it in its movement, and possess in consequence a considerable kinetic energy. This would appear if they were suddenly transported from on point on the globe’s surface to another endowed with a different velocity; for instance, from the pole to the equator. On arriving at the equator they would be hurled into space with a speed more than six times that of a railway train.

Independently of the movements of translation in a straight line like that of a cannon ball, or of rotation like that of the stars, matter and ether may show very different forms of movement. There result from this forces very different in aspect. We observe notably vibratory movements like those of a tuning fork, and circular disturbances such as those produced by casting a stone into the water, etc. Light and heat show exactly these last forms of movement. It is not only the kind of movements, but also the variations in velocity which condition the nature of forces. The recent theories on electricity put this last point well in evidence. They show, in fact, that forces differing from each other so widely as magnetism, the electric current, and light are generated by simple variations in the movements of electric particles.

An electrified body in repose produces effects of attraction and repulsion only, and possesses no magnetic property. Set it in motion, and it is immediately surrounded by magnetic lines of force, and produces all the effects of a current like that which traverses telegraph wires. Let us vary by a sudden acceleration the speed of the particles, and they immediately radiate through the ether. Hertzian waves, calorific waves, and lastly light. These forms of energy, although so different in kind, only appear therefore as the consequence of simple changes of movement.

The forces of nature probably contain other elements than movement. These elements do not affect our reagents, and we are therefore not cognizant of them. In the ocean of phenomena, science can only pick out what is accessible to it.

2. The Mechanical Explanations of the Universe

That which precedes makes us feel in advance how fragmentary, and consequently how insufficient, must be the final explanation of phenomena which the science of mechanics proposes.

Naturally this conclusion is not the one arrived at by the defenders of the doctrine which claims to explain everything by means of the equations of movement. In no way stopped by the excessive simplicity of their concepts, persuaded that all phenomena were wrapped up in their formulas, they have known neither mistrust nor uncertainty, and have imagined that they had for all eternity built up an edifice of imposing grandeur.

For the majority of scholars, this sublime confidence still endures. One of the most eminent among them, Cornu, the Academician, at the Congres de Physique in 1900, delivered himself as follows: —

“The spirit of Descartes soars over modern physics. What am I saying? He is its shining light! The more we penetrate into the knowledge of natural phenomena, the more developed and precise is the audacious Cartesian conception of the mechanism of the universe. There is in the physical world only matter and movement”.

At the very moment these words were uttered, the classic edifice was furrowed by deep chasms. While the mathematicians were drawing up formulas, the physicists were making experiments, and these experiments fitted in less and less with the formulas. These discrepancies, however, did not greatly trouble the mathematicians. So soon as the equations no longer agreed with the experiments, they rectified the equations by imagining the intervention of “hidden movements” which completely baffled observation. The process was evidently ingenious, but evidently also a little childish. “Since”, says M. Duhem, “no condition, no restriction, is imposed on these hidden movements, on what should we found the proof that a given difference may not find in them its raison d’etre?”.

Notwithstanding such subterfuges, the insufficiency of the classical mechanics has every day become more manifest with the progress of physics. “There exists”, writes the author I have just quoted, “a radical incompatibility between the mechanics of Lagrange”, that is to say, the classical mechanics, “and the laws of physics; this incompatibility attacks not only the laws of these phenomena in which the reduction to movement is the object of hypothesis, but also the laws which govern perceptible movements”.

It is not wholly in the great questions relating to the synthesis of the universe that the classical mechanics has shown itself very insufficient, but also in apparently much more modest problems like the theory of gases. It is by invoking the calculation of probabilities, by imagining a kind of statistics that it arrives at establishing extraordinarily complicated and also extraordinarily uncertain equations which elude all verification.

Professors who continue to teach the formulas of mechanics renounce more and more their belief in them. This fictitious universe, reduced to the points to which forces are applied, seems to them very chimerical. “There is not a single one of the principles of rational mechanics which is applicable to realities”, recently wrote to me one of the scholars who have most deeply sounded the problems of mechanics, the eminent Prof. Dwelshauwers Dery.

In fact, mechanics has fallen into a state of anarchy from which it does not seem likely to emerge, notwithstanding the numerous attempts made to transform it. At the present time there exist three very different systems of mechanics: —

  1. The classical mechanics, built up on the concepts of mass, force, space and time.
  2. The mechanics of Hertz, which discards the notion of force and replaces it by hidden links, supposed to exist between bodies.
  3. The energetic mechanics, founded on the principles of the conservation of energy, which we shall study later on. In this, matter and force disappear. There is not in the universe any other fundamental element but energy. This element is indestructible, while unceasingly changing its aspect. The various phenomena only represent mutations of energy.

We might, however, vary mechanical systems to infinity by replacing the concepts of time, space, and mass by arbitrary magnitudes and expressing phenomena as functions of these new magnitudes. This is sometimes done by introducing into the equations, instead of the coordinates of the classical mechanics, the physical magnitudes such as pressure, volume, temperature, electric charge, etc., which determine the state of the body. From the principles derived from the study of the dissociation of matter cited in a previous chapter, there might be deduced a new mechanics in which matter would figure as the source of the various forces of the universe. We should write in the equations that such and such a force is simply matter minus something, that inertia is a consequence of the relations of equilibrium between intra-atomic energy and the ether, etc. We should thus link force to matter, and we should express the former as a function of the latter conformably with the teachings of experiment.

But the moment has not arrived to translate into equations magnitudes of which the relations are not yet fixed. It is not very probable that this new mechanics would explain much better than the old one the mysteries of the universe.

The fact that we only perceive in the universe matter and movement does not authorize us to maintain that it is not composed of anything else. We can only say that by reason of the insufficiency of our senses and of our instruments we only perceive that which presents itself in the form of matter and movement. Twenty years ago, we might strictly have said that there was nothing else. But the very unforeseen phenomena revealed by the study of the dissociation of matter have proved that the universe is full of formidable powers hitherto unexpected, and has shown the existence of immense territories completely unexplored. The edifice built by science which has so long sheltered our uncertainty now appears like a fragile shelter, of which the entire foundations have to be set up anew.

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