Spencer Heath Archive

Item 347

Typed pages by Heath and two polished diagrams to have been submitted for publication to Glen W. Warner, Editor, School Science and Mathematics, 7633 Calumet Ave., Chicago 19, Ill., on recommendation of Lilly Rona. Whether that was done is not known. A penciled notation in Heath’s hand (but crossed out): “Must be re-written so as to take not mere motion but a rate of motion, velocity, as the middle term.” See also related Item 355.

1948

 

 

 

ENERGY, TIME, ACTION

 

 A CLARIFICATION      

     Physical science has proved a tremendous agency for rendering his habitat more habitable to man, thereby advancing his immortal dream of ever lengthening days. And science has no limitations but those of those who pursue and practice it. Rare discoveries from time to time have unearthed and formulated basic general principles in light of which whole worlds of past confusions have been swept away. Yet amid these triumphs of the seeking heart and seeing mind, may there not still linger, all unknown, fragments of old-time mists and mazes that confuse and lead astray? A case in point is our fundamental concepts of mass, motion and duration, of energy and time.

 

     Since adoption of the system of measures known as the absolute, there have been in existence two formulas for the measurement of energy, the older, dating from Galileo, based on gravitational force and therefore relative to particular gravitational intensities, and the modern scientific or absolute, based on inertial mass as invariable under all conditions.

 

     The older formula, based on gravitation, is

 

Energy = F x D

 

in which F represents a known weight (pounds) or equivalent force and D represents the distance (feet) through /which/ the weight or force is either able to move or through which it actually does move. In the one case the energy is called “potential,” in the other “kinetic” energy. In the one case the motion is hypothetical, hence no time is involved. /a line struck out?/

 

     The later formula, based on inertial mass, is

 

Energy = M x V²/2

 

in which M is the number of mass units in a moving body and V its velocity. Besides being based on an invariable mass or force unit, a further usefulness of this formula comes from its convenient application to any body of known mass M and known velocity V and thereby ascertaining, in terms of ergs (dyne-centimeters), what amount of energy, as conceived under the older formula in terms of force and distance, could have imparted the instant known velocity to the known mass, thus determining what amount of energy the moving mass must possess.

 

     Now there is a tendency to regard these two formulas as being in some respect fundamentally different. It seems to be assumed, even when not definitely asserted, that whereas in the simpler and earlier formula there is clearly no symbol for the involvement of time as a constituent or factor of energy, the same is not true of the absolute formula, and that this formula does involve time. Much color is given to this assumption by the presence in the latter formula of the letter V for velocity, thus seeming, at least, to involve time as an element in such energy as is measured or determined by it. Both formulas being at least empirically correct, if time is involved in the one and not in the other then there must be two kinds of fundamental energy. The confusion should be cleared up.

(The diagram on the accompanying sheet is to appear here with the following)

LEGEND:

Diagram showing the relation between time elapsed and velocity attained and distance in centimeters covered after one, two and three seconds by a gram mass accelerated by a constant rectilinear force of one dyne. The three divisions on base 0-N represent the successive elapsed seconds and attained velocities. The shaded unit-areas above these divisions — whole squares and half-squares — represent the amount of motion or distance due to initial velocity and that due to acceleration during each second. Wherefore M (V²/2) represents in actuality not mass into velocity but dynes into distance, and time is only inferentially if at all involved.

 

     The diagram above is a graphic representation of the derivation of the formula for energy, E = M(V²/2), in which the gram unit-mass M is accelerated by a constant force (dyne) sufficient to move it one-half centimeter in any one second and give it thereby and thereupon a velocity of one centimeter per second. For any number of seconds the elapsed time is represented by the equal intervals and the numbers set off on the base line 0-N. These lengths and numbers represent also the velocities in centimeters per second attained during that time.

 

     If on the time interval 0-1 we erect a unit square 0 1 Q R and let this unit of area represent one centimeter of motion or distance moved, then the half-square 0 1 Q can conveniently represent the half-centimeter that the unit mass moves during the first second in order to attain its final velocity of a whole centimeter per second. The distance so represented is a linear or spatial and not a temporal magnitude, notwithstanding that numerically it is the same as V²/2. Similarly, a square 1 2 V Q erected on the time interval 1-2 may represent the whole centimeter that the mass M moves in consequence of the velocity it has acquired at the beginning of this second time interval. The half-square Q V S then represents the half centimeter of distance gained by its acceleration during this second interval, the same as during the first interval, in which there was no initial velocity. The shaded square and half-square above the interval 1-2 thus represents the one and one-half centimeters of motion or distance moved during that second interval. And the total distance moved at the end of two seconds is represented by the sum of the shaded unit-areas above the total time interval 0-2. Here again the distance so represented is numerically the same as V²/2, although it is a spatial and not a temporal magnitude. The same is true at the end of any number of seconds. The half-squares or triangles above the lines represent the respective increases of distance moved due to the successive accelerations, while the whole squares below these triangles represent the successive increases of distance due to the successively increased velocities. Hence if we erect a square on any length of the line 0-N, which length marks the number of seconds elapsed and also the velocity attained, the half-square of the triangle formed by the diagonal from 0 will contain the same number of squares (whole squares and half squares together) as the number of centimeters that the unit-mass has moved during that elapsed time. It follows that for any given velocity, since the time elapsed is represented by the same number as the velocity attained, the total distance in centimeters moved is the mean velocity V/2 multiplied by the number of seconds elapsed, which is the same number as that of the velocity V. Hence the total distance (D in the earlier formula) can be taken indifferently as either mean velocity (V/2) x elapsed time (V), or (velocity² x V²)/2, or seconds²/2.

 

     Wherefore, in the formula, E = M(V²/2), M actually represents the number of dynes or force or inertia units, one for each unit of mass, and V²/2 represents in actuality the total distance through which such force must have acted in order to impart the given velocity.

 

     Therefore the two formulas for energy — force times distance or amount of motion, and mass times one-half the squared velocity — while they appear different in form are perfectly identical in fact. They denote energy as the product of force times motion or distance and nothing more. The erg, the unit of energy (not erg-second, which is action), is the product of only two factors — a force of one dyne into one centimeter of motion or distance. Energy is divided by time to obtain a rate, but energy does not include any factor of time. It is always potential, never more, because there is no action or actuality unless time is included as a third factor. Whenever time is admitted as a third factor the resulting actuality is action. And to no purpose can time as a factor be admitted more than once, for any attempt to admit time as a fourth factor or dimension could do no more than increase the magnitude of the time factor already employed.

 

     Metaphysics is wholly conceptual. It may or may not have any counterpart in that objective side of experience which is called action or actuality and constitutes this reality which physical science investigates and with which, by its measurements and ratios, physical science rationally deals. The field of science is the objective world of acts, happenings, events; of actuality as action. It finds that this objective and concrete reality possesses just three fundamental elements or aspects which together constitute any action or event.

 

     These three constituent aspects of action, mass, motion and time, are mutually involved and inseparable in any event. Events may be examined and conceived in any one or two of their fundamental aspects by disregard of both those or of the one remaining, but they are to be experienced as actuality only when combined in the three-fold unity of a happening or event.

 

     Notwithstanding this inviolable integrity of events, science can and does measure and examine them in terms of their three separate and distinguishable aspects. To this end there have been devised three fundamental units of measurement. There is a unit for taking the dimension of mass as inertia or force; that unit is the dyne. There is a unit of motion or distance; that unit is the centimeter. And there is a unit of time or duration; that unit is the second. The numbers of the units contained in any magnitudes of these several kinds are the dimensions of these three kinds of magnitudes — the fundamental magnitudes of physical science, whence all its other magnitudes are compounded or derived and all its dimensions assigned. If and when any fourth form or element or aspect of actuality becomes manifest as being involved in the objective side of experience and if or when a common standard unit is devised for the measurement of such fourth element or aspect, it will then and only then be possible to take dimensions of such fourth fundamental unit. Dimensions of any one kind may be repeated indefinitely; but four kinds of dimensions cannot be taken unless and until there are more than three kinds of fundamental units in which to take them.

 

     But the anomaly of two kinds of formulas for energy is more apparent than real, for in the practical application of either formula the actual procedure is the same. Remembering that corresponding symbols (as we have seen) have the same meaning in both formulas — that M and V²/2 in the second stands for precisely the same as F and D in the first, let us examine just what their significance is, employing F and D as applicable to both.

 

     We will find that our formula indicates only the first step towards determination of the over-all magnitude of an event — the quantity or amount of action involved in the unity of its three elemental dimensions of mass, motion and time. The letter F seems to stand for a number of force units without any reference to motion or length, and D seems to stand for distance or length without reference to time. And no symbol for the third necessary magnitude, time, appears explicitly at all.

 

     What F really implies is a given number of force units per unit of motion or distance. It is really a ratio, whose denominator is one — one unit of length. It is force units stated in terms of a length unit, pounds with regard to one foot, pounds per foot, which is a ratio and not a mere quantity. The letter D, however, represents a quantity of motion and not a rate. F x D then is a force-distance ratio multiplied by distance, F/1  x  D, whichever formula is used. The E in both stands only for a quantity of work or the potential (imaginary) energy to perform it. Before E can represent even a ratio of work to time, the motion or distance must be divided by its inevitable time, to get the rate of motion, which, multiplied by the force ratio, gives the energy rate. Then, to ascertain the quantity of energy flow during some inevitable period of time, this energy rate must be multiplied by the number of time units contained in that period. This gives the amount of energy in action or, more simply, the quantity of action thus involved. It is only after motion or distance has been turned into a rate by dividing distance by time that E can represent an energy rate. And it is only when this rate has been multiplied by some quantity of time that E can represent any amount of energy in action — a quantity of action, such as so many pounds-feet-minutes or so many of the dynes-centimeters-seconds that we call erg-seconds.

 

     Now a number of force units cannot be multiplied by a number of length units any more than a number of horses can be multiplied by a number of apples. However, if horses and apples are coming and going in such proportion that the constant ratio of horses to apples is, say, thirty to one, that is, there are thirty horses every time there is one apple, then apples can take the place of times. The frequency of horses in terms (times) of apples is thirty and 30/1 expresses the ratio of horses to apples. This /is/ a heterogeneous ratio in that it relates a quantity of one kind to the unit quantity of a different kind. The thirty horses are not divided up among the apples; there are thirty horses each time there is an apple, just as there would be thirty pounds for each foot if pounds and feet instead of horses and apples were being illustratively employed.

 

     If now we have twenty apples, this number, 20, is our apple dimension, for it expresses the ratio, 20/1, of a quantity to a unit. And since the unit is of the same kind, it is a homogeneous ratio. (All dimensions are homogeneous ratios.) Since any ratio is a number of times or counts, we can multiply the apples-to-apple ratio, 20/1, with the horses-to-apples ratio, 30/1, without multiplying either horses by apples or apples by horses. For our multiplier, seen in this light, is a number of times and not any number of either apples or horses, and our product is 600 horse-apples, just as it would have been 600 pounds-feet had those two kinds of quantities been employed instead.

 

     The foregoing illustration would have been precisely the same if pounds and feet instead of horses and apples had been employed throughout. When we do this we in reality multiply the recurring dimension of feet with the heterogeneous ratio of pounds to feet, namely 30/1, and our new combined unit, pounds-feet, has the magnitude of 600 which is its dimension because it is, in terms of its own unit, a homogeneous ratio of 600/1.

 

     It is important to reflect that this work dimension, 600 pounds-feet, is merely a ratio — a quantitative relationship to its own unit, the pound-foot — and that as mass times motion alone it does not possess any actuality as action, experience or event. Unless or until the necessary third factor, time, as duration or frequency, is admitted, it is wholly abstract and subjective, just as any abstract quantity is wholly subjective unless it is attached to some object or event that can be experienced. So, in order to bring our conception of 600 pounds-feet a step nearer to the world of actuality we must stipulate not only the force and motion or distance involved but also some rate per unit of time at which the motion takes place.

 

     Let us say that the time required for each twenty feet of motion or distance is one-fifth of a minute. Then the velocity is 20/(1/5) or 100 feet per minute. The work rate is now 30 x 100 or 3000 pounds-feet per minute for whatever number of minutes (if any) that it may continue. We have now associated with mass and motion a third abstraction, namely time, and by relating time to distance or motion have interposed a second heterogeneous ratio, namely acceleration. These two ratios are mass (force) in terms of motion or distance (thirty pounds per each foot) and motion or distance in terms of time (twenty feet per each one-fifth minute). Combining these two ratios: force in terms of motion units (30 pounds to each foot), and motion in terms of time units (20 feet to each 1/5 minute), we obtained a third heterogeneous ratio or dimension, a rate of energy in terms of time units, 30 x 20 pounds-feet to each 1/5 minute, or 3000 pounds-feet per minute.

 

     But even so, we have not yet introduced time as a quantity in combination with force and motion — an element that is essential and inevitable in any action or event that is objective and not purely imaginative. We have employed one-fifth of a minute only as a unit of measure for dividing off quantities of motion. We have divided motion by time and thereby obtained only the ratio of motion to time — what quantity of motion is rationed to a single unit of time. By this division we have established a rate or ratio at which our mass or force moves through a given distance per unit of time. Our 600 pounds-feet has become 3000 pounds-feet per minute. And this, too, is hypothetical. It is not now the distance through which, but the rate at which, the stipulated (or measured) force moves. But if it does so move it must do so during some period of time. It is the dimension of this period, its duration, that transforms the abstract rate of action (commonly called energy) into the concrete reality of three-dimensional action.

 

     To bring our formula into the world of objective reality we must incorporate in it a quantity or duration of time, say, ten minutes. The symbol E can then represent not merely a static concept of mass or force and length, not merely the kinetic concept of an abstract rate or ratio, but in addition to these all the actuality of a definite concrete event. E can now represent 3000 pounds-feet per minute acting for, say, ten minutes. Thus 50,000 pounds-feet-minutes is a fully defined action or event in that it defines an event completely in terms of the three always essential elements or aspects, mass, motion and time.

 

     Returning now to our two formulas for energy, as amended by the inclusion of time to give them the objective significance of energy in action, we will substitute, first in one and then in the other of them, the same numerical values as heretofore, while retaining in each formula the basic units peculiar to it.

 

     In the formula, E   F x D x T,    Force, Distance, Time,

 

E  is  Energy in Action, pounds-feet-minutes.

F is  30 pounds per each foot of motion or distance.

D is  20 feet per 1/5 minute.

Frequency is  5 per minute, the frequency of the 20 feet.

Velocity  is  5 times 20, 100 feet per minute. This refers to uniform          velocity.

F x D is  600 pounds-feet of work or potential energy.

F x 100 is  the energy rate, 3000 pounds-feet per minute.

T is  10 minutes, duration of the energy at the given rate.

E  is  30 x 20 x 5 x 10, or 30,000 pounds-feet-minutes of Action.

 

In the formula, E = M x (V²/2) x T, corresponding with F x D x T in the other,

    E  is  Energy in Action, dynes-centimeters-seconds.

    M  is 30 dynes for each centimeter of distance

   V²/2  is 20 centimeters per 1/5 second. This is not velocity squared but velocity, 100, x time, 1/5 second, the same as V²/2 in the formula, where it is really the average or uniform velocity x time and is thus a distance, 20 centimeters, the same as D stands for 20 feet in the other formula.

  Frequency  is 5 per second, the frequency of the 20 centimeters

   Velocity  is 5 times 20, 100 centimeters per second.

    M x V²/2  is  30 dynes times 20 centimeters, 600 dynes-centimeters.

   M x 100  is the energy rate, 3,000 dynes-centimeters per second.

         T  is 10 seconds, duration of the energy at the given rate.

          E  is 30 x 20 x 5 x 10, or 30,000 dynes-centimeters-seconds     of Action.

 

     From the above it should be clear that the two formulas although differing as to their symbols are nonetheless identical in fact. But unless supplemented by an inclusion of time as above both are void of any objective meaning. In the older formula, E is simply force times distance or motion, without any symbol for either a rate of motion or a duration. The later formula suffers the same omissions, but they are obscured by distance V²/2 being stated in a symbol strongly suggestive of velocity and therefore of time, thus making a false appearance of time being a factor in the formula. To give E any significance as a rate of energy, it is necessary to assign to the motion or length quantities, D and V²/2, a measure of time. When this measure is divided (not multiplied) into the measure of length we have the rate or velocity with which the force moves. Then, to give E any significance as a quantity of energy, we must introduce in each case a period of time, whereupon E in either case represents a quantity of energy acting at a rate of and through a period of time. E then stands as units of Action — force times velocity of motion times its duration. — Or, force times a length of motion times the frequency of the motion times its duration.

 

     This full formula for energy in action has the same form as that of the celebrated quantum of action discovered to the world in 1900 by Professor Max Planck. It is a rate of motion, velocity of a force, multiplied by its duration. The only difference is that in the Planckian formulation the product of force times velocity, or force times its motion times its frequency of motion, multiplied by the duration can never be greater nor less than a very definite but almost infinitesimal fraction of an erg-second (dyne-centimeter-second). It seems to be a law of nature that energy as Action never manifests itself in any smaller magnitude and that all higher magnitudes shall be even multiples of this least. So, we may say that all actions or physical events that can be experienced are almost infinitely complex whole multiples of that least unit of action or event which is symbolized as h and has a magnitude of 6.5 x 10^-27 erg-seconds.

 

     Figure 2 shows diagrammatically the three-fold character of this “atom of action.”

 

     As the single hydrogen atom is fundamental to all atomic structures, so does this irreducible unit of action seem fundamental to the structure of all possible physical events and a clue to the rational analysis and understanding of them. It is of interest to note that in these least units of action there are united in many different proportions always just three constituent elements. There is an element of mass, force, inertia; an element of motion, distance, length associated with a unit of time as rate or velocity; and there is an element of duration, of time units in rhythm or repetition. If the product of these three, in whatever proportions, must have always the same over-all magnitude, it follows that when one is at its minimum the other two, taken together, must be maximum. For example, if the mass, force or inertia element (moving particle) is at its minimum, then motion per unit of time (velocity) must be maximum; if motion is at its minimum, least possible velocity, then mass, force or inertia must be at its maximum; and if time be at its minimum, then force per unit of motion or distance must be maximum. All this suggests a rational frame of reference for atomic explosion and radiation on the one hand, and maximum mass, with least motion and temperature and maximum entropy and density on the other hand, and it accounts for the limited constant velocity of radiation, including light.

 

     A dimension is the number that expresses the ratio between any magnitude and a unit quantity of the same kind. Thus all dimensions are ratios, numerical relationships, abstractions. There can be as many dimensions of any one kind as there are magnitudes of that kind, but there can be no more kinds of dimensions than there are kinds of units and magnitudes.

 

     However, there may be imaginary magnitudes not capable of being experienced because lacking one or more of the three elements necessary to constitute action or an event. Such is any quantitative conception of either mass or of motion or of time, when imagined separately without reference to the other two. So also are there composite imaginary magnitudes, such as those composed of mass (as force or inertia) and motion. The unit of such magnitudes is the erg, and their dimensions are their numerical ratios to this composite unit. And again, magnitudes composed abstractly of motion or distance alone, and as in two, three or more di­rections, but without any units of mass or of time, are the abstractions of surfaces and volumes. All these are only formal and empty representations — abstractions away and apart from the actuality of any concrete unity with mass or inertia and time. Their only dimensional unit is the centimeter or other unit of length, and their dimensions are their numerical ratios to this unit. So also there are magnitudes composed abstractly of motion or length and time; such is the concept of waves. These are only formal and empty representations of action — abstractions of length and time away from particle, inertia or mass. Their dimensional units (as yet unnamed except as space-time) are those of length or motion and duration or time. The unit of wave magnitude, then, is the centimeter-second. And the over-all dimension of any wave, considered as devoid of mass, is the number of times this unit is contained in it; and its velocity is its length times its frequency, which is constant for all waves.

 

     But let none be confused or deceived. Not one of these partial, abstract and imaginary integrations rises to the level of actual objective or experiential reality. Only when we in our minds integrate the three full fundamentals of physical science, the same as do our objective experiences, can our conceptions rise out of the mystical and abstract to the level of concrete reality. Any integration of mass (as force or inertia), motion (as length or distance), and time (as rhythm or repetition) is an action or event. Events are actualities. Action is reality. In this reality, the concrete world presents to our experience nothing but such fully integrated events. Only so far as our conceptions are likewise fully integrated do our minds dwell in the realm of reality. We can take concrete events apart imaginatively, conceive their elements separately in terms of their several dimensions, and they can be transformed, both imaginatively and actually, into changed or new concrete events by rearranging the ratios and the magnitudes in which their three constituent elements are combined.

 

     This purposeful re-proportioning of the mass, motion and time in events is the basic process that underlies all the rational technologies that serve mankind. And here it is that science lays into the hand of man creative power — the gift to build, in present fact and not in fantasy or faith alone, his hope and deathless dream.

 

 

[Two pages of polished diagrams accompany this.]