/Of five pages, typed and numbered by Heath, number 4 is missing./

 

… of length, and their dimensions are their numerical ratios to this unit. So also are there magnitudes composed abstractly of motion or distance and time, such as 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. The dimension of any wave is the number of times this unit is contained in it, and its velocity is its length times its frequency, which is a 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 full three 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 into action of mass (as force or inertia), motion (as length or distance), and time (as repetition) is an action or event. Events are actualities. Action is reality. And in this reality the concrete world presents to our experience in ac­tuality nothing but 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. And this reproportioning of mass, motion and time is the basic process into which all technologies resolve. 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.

 

/Here end the pages typed by Heath. Following below are transcriptions of pencil notes accompanying these pages and presumably to have been inserted somewhere above./

 

 

 

 

 

 

[“Insert all this on page 4”]

 

 The anomaly between the two formulas for energy is more apparent than real, for in the practical application of either formula the actual procedure is the same. (*) Both take only the first step towards determination of the over-all magnitude of an event — the quantity or amount of action involved within its three elemental dimensions of mass, motion and time. The older formula contains a symbol for the first of them F in terms of gravitational “force” without reference (or involvement of) distance. The second symbol it gives in terms of distance without reference to or involvement of time. And it offers no symbol for the third necessary magnitude, that of time.

 

 But what F really implies is the number of pounds per unit of distance. It is really a ratio whose denominator is 1. It is force stated in terms of motion, pounds with regard to one foot, pounds per foot, which is a ratio and not a quantity. 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, a quantity of work, whichever formula is used. The E in both cases stands only for a quantity of work or the potential energy to perform it. Before it can represent a rate of work to time, the distance or motion must be divided by its inevitable time to get the rate of motion, or velocity, which multiplied by the force gives the energy rate. Then to ascertain the quantity of energy flow during any period of time this energy rate must be multiplied by the number of time units in that period. This gives the amount of energy in action or, more simply, the quantity of action during that time.

 

 It is only after motion or distance has been turned into a ratio 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 foot-pound-minutes or so many 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 should be thirty pounds for each foot if pounds and feet instead of horses and apples were being illustratively employed.

 

 If now we use 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 a ratio is a number of times, we can multiply the apples-to-apple dimension, 20/1 (homogeneous ratio), with the horses-to-apple ratio, 30/1 (heterogeneous ratio), without multiplying either horses by apples or apples by horses; for our multiplier, seen in this light, is a number of times and not a number of horses, and our product is 600 horse-apples (just as it would have been 600 pound-feet had these two kinds of quantities been employed).

 

 The foregoing illustration would have been precisely the same

had pounds and feet instead of horses and apples been employed. When we do this, we in reality multiply the recurring dimension of feet by the heterogeneous ratio of pounds to feet, namely 30/1, and our new combined unit, namely, 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.

 

/// — 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.

 

 Combining these two ratios we obtained the new kind of ratio or dimension in pounds-feet, and by stipulating its frequency as 5 per minute (its period being 1/5 more /?/) we have established a third heterogeneous ratio, pounds-feet per unit of time, (30 x 20 x 5)/1. But this far we have not introduced time as a quantity. ///

 

 It is important to reflect that this work dimension, 600 pounds-feet, is merely a quantitative relationship, a ratio. Further, it is a relationship between quantities to its own unit, pounds-foot, mass and motion combined — as which, taken alone, there is no 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 of work 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 20 feet of motion or distance is one fifth of a minute. Then the velocity is 20/5 = 100 feet per minute. The work rate is now 30 x 100 = 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 have introduced a second heterogeneous ratio, namely, acceleration. These two ratios are mass (force) in terms of motion or distance (30 pounds in terms of _______) and motion or distance in terms of time (20 feet in 1/5- minute rate = 100 feet per minute).

 

 When these two heterogeneous ratios are combined we have the new kind of ratio, pounds-feet-per-minute, represented by 3000 pounds-feet-minute in the homogeneous ratio of 300/1 as the dimension in terms of its own kind of unit, the pound-foot-minute. But thus far we have not introduced time as a quantity but only as a ratio. In the ratio that we call acceleration we have employed one-fifth minute only as a measuring stick for marking off quantities of motion in terms of time. We have divided motion by time. But the quantity divided is motion or distance and not time.  [Triangle mark] And by this division we have established a rate or ratio at which our mass or force moves through the given distance per unit of time. Our 600 pounds-feet have become 3000 pounds-feet per minute. And this too is hypothetical. It is not now the distance but the rate at which the hypothetical mass moves (work is performed) if there is any actuality or action — if the hypothetical mass or force actually does move through the hypothetical distance during a hypothetical time.

 

 If it does so move, it must do so for some period of time. It is the dimension (numerical magnitude) of this period that transforms the abstract rate of action (that we commonly call energy) into an actuality as a quantity 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. Our symbol E can then represent not merely a static concept of mass and length, not merely the kinetic concept of an abstract rate or ratio, but in addition to these all the actuality of a definite objective event. E can now represent 3000 pounds-feet per minute acting for say 10 minutes. This is a fully defined action or event as energy in action, at the rate of 3000 pounds-feet per minute for 10 minutes — because it defines the event completely in terms of its three fundamental aspects, mass, motion and time.

 

 To return now to our two formulas for energy, let us substitute the numerical values we have been using. In the older formula

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This purposeful reproportioning 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.

 

[This suggests an essay on the absolute TRINITY.]

 

when combined in their threefold unity as a happening or event.

 

_____________________________

[For top of page 4]

 

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

 

 The three aspects of action, mass, motion and time, are inseparable and mutually involved in any event. Events may be examined and conceived in any one or two of their three fundamental aspects by disregard of the remainder but they are to be experienced as action or actuality only in their entire unity as happening or event.

 

 Notwithstanding this inviolable integrity of events, science can and does examine and measure them in terms of their three separate and distinguishable aspects. To this end there have been devised three fundamental units of measurement. There is .. [Continue as at top of p.4.]

__________________________________

Combining the two ratios — mass per unit of motion and motion per unit of time — we obtained a new kind of ratio or dimension in pounds-feet and by stipulating its frequency as 5 per minute (its period being 1/5 minute) we have established a third heterogeneous ratio, pounds-feet per unit of time, (300x20x5)/1 or 300 pounds-feet per minute. But, even so, we have not yet introduced time as an actual quantity. We have analyzed 1/5 minute only as a . . .

 

 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 for some period of time. It is the dimension (numerical magnitude) of this period that transforms this abstract rate of action (commonly called energy) into the concrete reality of three-dimensional action.

 

• – – – – – – – – – – – – –

 Combining the two dimensions, force in terms of motion units (feet) and motion in terms of its own units (feet), we obtained a new dimension, work, in terms of its own units (pound-feet) and stipulating its period as 1/5 minute or the frequency, 5, we have established a third heterogeneous ratio, pounds-feet per unit of time, (30x20x5)/1 or 3000 pounds-feet per minute.

 

 But, even so, we have not yet established time as a quantity in combination with mass and motion. We have employed one-fifth of a minute only as a unit of measure for making up quantities of motion. We have divided motion by time and thereby obtained only the ratio of motion to time — what quantity of motion is related to a single unit of time. But the quantity divided is motion or distance and not 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.

__________________________________

 

 Figure 2 shows diagrammatically the two-fold character of this “atom of action.” As the single hydrogen atom is fundamental to all the atomic structures so this individual unit of action seems fundamental to the structure of all possible events and a clue to rational analysis and understanding of them. It is of interest to note that in these least units of action there are united in varying proportions always just three constituent elements. There is an element of mass, force or inertia; an element of motion, distance or length associated with a unit of time as rate or velocity; and an element of duration —time units in repetition. If the product of these three has always the same over-all magnitude the it follows that when one is at its minimum the others, taken together, must be maximum.  [Continue as per page 18]

 

 

 

 

 

 

 

 

[* Insert on page 2]

 

If on the time interval we create a unit square O-I-Q-R and let this unit of area represent one centimeter of motion or distance, then the shaded half-square O-I-Q can conveniently represent the half-centimeter that the unit mass M moves during its first second in order to attain its final velocity of a whole centimeter per second. This distance is a linear or special and not a temporal magnitude notwithstanding that numerically it is the same as V²/2. Similarly, a square I-2-V-Q erected on the time interval I-2 may represent the whole centimeter that the mass M moves due to the velocity it possesses at the beginning of this second time interval. The half-square Q-V-S then represents the half-centimeter of distance gained by acceleration during this second interval, the same as during the first interval, in which there was no initial velocity. The ______ square and half-square above the interval I-2 thus represent the one and one-half centimeters of motion or distance moved during that second interval, and the total distance moved at the end of ten /?/ second is represented by the total of shaded unit areas above the total time interval O-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 at the end … etc.