# 18: How to Measure Variations

We now need to examine more closely our two ways of assessing biological entities; their internal energies; and the populations they form. Those two ways are obviously the key to separating what they must all do from what they must each do.

We can see our two methods in action in the picture we took (Figure 18.1) of our experiment with Brassica rapa. We can use it to relate the two methods together very easily:

1. Our first method is that of fluxes. This handles the whole population—i.e. the complete simultaneous congregation of internal energy—and is what they must all do.
2. Our second method is that of tensors. This handles each entity separately—i.e. each one's distinct partition of internal energy—and is what they must each do.

Our first method, with fluxes, treats the population as a whole. We always measure the collected assembly of all plants together, in all the pots. We record the entire Mendel and Wallace pressures, or the mass and energy fluxes, M and P. This is the complete stock of internal energy the population uses. We call the resulting period of time T. It is a definite amount of elapsed clock time of t seconds.

So far so good.

Our second method, with tensors, treats the plants individually. We assess each plant individually, each from its own point of view. We carefully count and measure each one. We record the numbers, n, in the population at each point along with their average individual mass, , and their average individual energy, , which is each of their separate stocks of internal energy. It is the contribution each one makes.

We keep counting and measuring the population and its internal energy until we observe our values beginning to repeat. We have then encompassed the entire set of activities that make up the temporal distribution, τ. That entire set is the sum of all the infinitesimal increments in internal energy all about the generation, which is ∫ dτ. We then call that interval “the generation length”. This is a definition.

The generation length is the sum of all the infinitesimal increments in internal energy over time, and that make up a generation. This introduces the third anomaly, for we now have T = ∫ dτ or similar. But this also raises the possibility that dt = Tdτ, or something similar; i.e. that every moment of time, dt, that passes is measured by the amount, dτ, traversed along some specific generation length, T.

We seem now to have two different ways of measuring the circulation of the generations. We can measure it either as an amount of elapsed clock time, T, or else as a specifiied sequence of events, τ. An infinitesimal stretch of the former is either dt or dT, that of the latter dτ

We are now ready to apply Newton's infinitesimals.We will apply them to our two methods of reckoning the generation length: T and τ. A generation is both the collected set of activities that make up a generation, τ; and a linear sequence of time of a set of durations, t. We then want to see if their two infinitesimals—dt and dτ—increment in the same way.

Our two methods of reckoning the total fluxes in internal energy might produce the same results, but that does not mean they are the same at every point. Clearly, what the members in the population must all do need not be the same as what they must each do. The problem is distinguishing these in a systematic way.

1. The method of fluxes measures the two population internal energy fluxes of M and P over a long series of infinitesimal increments each of length dT. We have infinitesimal increments of time the population accumulates, irrespective of their numbers. Since we want to infinitesimally sum all those increments in the fluxes over each infinitesimal period of time that we measure them, this give us the grand totals of ∫M dT kilogrammes for the mass, and ∫P dT joules for the energy. This is what they must all do.
2. The method of tensors instead gives us a count of dn members in the population in every infinitesimal increment of time. We have infinitesimal increments in numbers. We also measure their changes in internal energy as an average individual mass of dm̅ and as an average individual energy of dp̅ on each occasion. We want to sum all those infinitesimal increments in number, and each of their infinitesimal increments in mass and in energy. This alternative method of approach gives the grand totals of ∫dm̅ dn kilogrammes for the mass, and ∫dp̅ dn joules for the energy. This is what they must each do.

Since “things that are equal to the same thing are also equal to each other”, we can put the two methods side-by-side. Ostensibly, we have the two relations ∫M dT = ∫dm̅ dn for the mechanical chemical energy aspect of their internal energy; and ∫P dT = ∫dp̅ dn for the nonmechanical chemical energy aspect of that same internal energy.

We must be very careful, however, about making any unwarranted assumptions concerning the equality, even identity, of these two sets of values and methods. One set of values—∫M dT and ∫P dT—involves increments in time, but is independent of what happens to distinct population numbers. The other set of values—∫dm̅ dn and dp̅ dninvolves increments in numbers and properties, but is irrespective of the times taken over those changes within the circulation.

We obviously get the same totals either way. But since one total for the mechanical chemical energy or mass is ∫M dT and so is independent of changes in numbers, while the other total is ∫dm̅ dn and so is independent of the times taken, there is no guarantee that they will increment in the same way. There is a distinct and logical possibility that there could easily be variations in the ways in which numbers and their partitions increment over time. That is to say, the infinitesimal increments in number, which is the number of ways the internal energy is partitioned, do not have to be equal to the infinitesimal increments in time for that same internal energy. The same holds for the ∫P dT and ∫dp̅ dn which are the nonmechanical chemical energy values. Those proposed equalities between the respective totals depend upon how numbers change with time and are instead what we must investigate and prove, and not what we should assume.

Tensors are used in exactly these kinds of situations.

Since we want to use our tensors to conduct a thorough investigation, we can fill the elements with whatever rules, values, and transformations we deem necessary. So we could easily fill one element or component in our tensor with the rule “compare x to x so as to produce 1”. We could certainly write this as x ÷ x = 1, and call it division. It is a way of producing a basis for making a comparison. It is how we create our orthonomal basis.

Once we have our orthonomal basis, we can compare the number of partitions in our internal energy stocks at different times. We are then comparing the number of distinct biological entities at each moment.

The size of our basis or unit of measure of course changes depending on which of any two phenomena we choose to measure first. The value we get for the comparison always depends upon which we choose to measure first. However, the value will also always be some proportion of the basis. So if we measure A; get a basis of 1; measure B; and get a value of 2; then if we switch things around and instead take B as our basis and then measure A, our measurement instead becomes ½ or 0.5. So our relative difference is expressed as twice as much if we measure one way, but as half as much if we measure the other. The total value of ‘basis-and-measurement-made’ remains the same in both directions. Only the numbers change.

We can infinitesimally increment, and compare again, taking both the new and the old measures, separately and together. That would give us the increment, but in terms of our basis or reference value potentially twice over. We would get it once for each direction we do it. This indifference to directions of measuring is again the beauty and simplicity of tensors. It also means that we can go out into the real world and deliberately measure in both directions. We can then compare the two basis-plus-measure sets. Once we have measured in whatever directions we can examine the values to determine whether or not things really are the same; and if things are not the same then we can ascertain the basis for whatever differences there might be.

Since tensors are ultimately indifferent to the direction of measurement, each rule for a value or component will have its inverse rule that allows us to reverse any transformation. So if we take the 1 we just created in the immediately above x ÷ x = 1 division process and do the inverse, which we can call multiplication, then the original is restored as in x × 1 = x. We then know the comparisons and transformations are valid. We are free to measure either way, and take whichever one we want randomly as our basis. If it is not 2 then it is ½, which ends up the same, because if we get ½, then we will get the 2 if we turn round and go in the opposite direction; and if we get 2 and turn around, we will get ½.

This ability to produce the same result either way is called an “identity operation”. And where there is an identity operation, there is an “identity element” to facilitate it. In this multiplication-division case the identity element is 1 because we have x × 1 = x, where we get x back again by using 1; and x ÷ x = 1 where we produce the 1 by the converse process of setting x against itself. We get 1 if we move in both directions, and whether we multiply or divide. We therefore know that 1 is the “multiplicative identity”, or “multiplicative element”. Whichever value we take first, and then divide, or else multiply, to get the other, we will get the same values both times. If our numbers end up the same in both directions, there is no variation in what we are measuring. There is no variation because we have worked with, and through, the identity element.

There is another very obvious way to make a comparison. If we now ask what the difference is between y and y, the answer is ‘zero difference’. We could just as easily write it as y - y = 0, and call it subtraction. We can then infinitesimally increment, and difference again to see what we get.

But the differencing operation must also be indifferent to which way we use it. If A is ‘2 more than B’, then B should also be ‘2 less than A’. If it is not, then something strange is happening to the thing we are measuring. So if we want the full freedom to take either as a basis, then we must be able to take any resulting zero difference and reapply it to end up with the original as in y + 0 = y. It now doesn't matter which way round we measure, and this is again an identity operation. We now have zero as the “additive identity” or “additive element”.

Tensors can be as simple or as complex as we wish. We can even examine the same values, but using different methods in different positions or components, if we so desire.

Tensors can also of course contain infinitesimals, meaning differentials, derivatives, and integrals. They do not have to. But they can … which is a little bit more tricky. The rules regarding integration and differentiation are considerably more complex, but if a tensor contains a rule that says integrating something is true, then since it is the inverse process, the element within the tensor must usually obey the reverse rule saying that differentiating that same element is also true and produces the same value, so that we restore the original. If A grows at a certain pace to produce B, then we must be able to diminish B at the inverse rate to produce A. There will again be an identity element relative to this process.

A mathematical aside

We need to get the same value whether we integrate or differentiate x, which was something discovered by the Swiss mathematician Leonhard Euler. Thanks to him, we now know that ∫ex dx = ex, and that d(ex)/dx = ex, with e being the identity element for integration and differentiation.

Tensors do not have to contain infinitesimals within their elements. However, we want to follow biological populations around the entire circulation of the generations so they are enormously useful. We will place all those changing population values in a tensor so we can compare them to each other. Since we will be infinitesimally manipulating entire tensors, their elements and components must obviously follow the relevant integration and differentiation rules. They must respect those two sets of procedures throughout that entire process. Tensors are not themselves infinitesimals, but they must facilitate and follow all relevant rules so we can take our measurements of growing plants and their internal energies, and then compare them.

We must make it easy to create a tensor that incorporates the transformations that recreate a population's values at any point in terms of its values at some other selected point. Any one population can then be our standard of measure relative to any other, and so that we can determine incremental and infinitesimal transformations between one point, or plant, or population, or generation and another in whatever manner.

Figure 18.2 Figure 18.3 Figures 18.2 and 18.3 give a visual representation of our two methods of fluxes and tensors. We are looking to measure a population and its internal energy as it is changing in time. We are using the two different methods at our disposal: one based on the t of absolute clock time; the other based on the τ of relative biological-generational time. Both of our methods use a planimeter, an instrument for measuring areas. Planimeters can be used either to circle regions on maps, or else in real-life surveying situations where they draw the boundaries of regions on the ground. Either way, they give us an area by measuring around the boundary.

Figure 18.2 is the method used by a ‘linear planimeter’. This gives us the total values ∫M dT and ∫P dT. It can trace all around an object's boundary and measure its length. We get the leaf area, which is in two dimensions, simply by measuring its boundary, which also moves about in two dimensions. That boundary may ostensibly be a one-dimensional linear measure, but it is procured by chasing up and down and in and out around the entire boundary. The linear planimeter converts that one-dimensional measure into a two-dimensional value for the area it has bounded. It determines that area from the infinitesimal changes it goes through in x and y as it tracks about and changes in first one dimension, x, and then in the other, y. It thus sets one linear measure against another, multiplying them together to produce an xy for the area.

The linear planimeter's ability to convert a line into an area is extremely useful. So we'll put one of those in our tensor. It beautifully combines our processes of (1) addition-subtration; (2) multiplication-division; and (3) integration-differentiation. It therefore allows us to make the kinds of comparisons we need, but all in one place.

Let us now carefully note … physical space gives us the three dimensions x, y, and z. Our biological space also has three dimensions. Our three biological dimensions are numbers, mass, and energy, or n, m, and p. In both the spaces, a journey around the boundary is going to take a very definite amount of time, and the planimeter's tracer will move at a very definite speed. It is always going to take a given amount of time to cover any kind of distance along each of these three different sets of axes. We can therefore determine rates for both lines and areas. We get the boundary of a physical interactive area in one case; and the boundary of a biological generation in the other.

So … now … the length of that boundary in our biological space has become an exploration of mass and numbers in time. The length of that boundary is now our generation length all about the time span of a generation. It is measured either as the complete boundary length T, or else as ∫dτ, with this latter being the infinitesimal sum of the discrete set of temporal distributions, τ, we will pass through.

And … since we are measuring in a biological space, then instead of the x and y of physical space, those two axes that the planimeter is recording have become m and n, which is the mechanical chemical energy of each biological entity juxtaposed with the numbers in our population. So instead of measuring an area as xy, our linear planimeter is now determining the complete mass of resources the population uses over that interval as a part of its internal energy. Thanks to our linear planimeter, therefore, we now have our ∫M dT kilogrammes of the mechanical chemical energy aspect of internal energy.

We can then do exactly the same for the other two dimensions of energy and number, and get our ∫P dT joules of the nonmechanical chemical energy aspect of the same internal energy, and in that same generation enclosed within our boundary.

Figure 18.3 then shows the process with the very similar ‘polar planimeter’. It gives us ∫dm̅ dn and ∫dp̅ dn. It measures the angle through which the polar planimeter is rotated. It traces the boundary as the linear planimeter did, but for a different reason. The angle is measured as the tangent, which is yx. If either x or y changes, then the angle changes. The polar planimeter links this angle measured to the radius it is simultaneously determining. As we saw in Figures 0.3 and 0.6 in Before We Begin, it then uses the radius and the angle to produce the area of a matching circle.

So where the linear planimeter measures the total flux—i.e. what they must all do—by only tracing a boundary length; the polar one measures an average radius—i.e. what they must each do—along with a net angular displacement or relative time spent on given activities. It thereby produces the same value but using a different method.

So … let's shove one of those very useful polar planimeters in our tensor, as well.

Although the linear and the polar planimeters produce the same overall total, they use different methods. For one thing, the linear planimeter works with xy and so tends to have a linear velocity directly along and about its working perimeter; whereas the polar planimeter works with the tangent of its angle as yx and so tends to have an angular velocity all about the circulation of the generations. The two have … and they measure … important variations.

Since the linear and the polar planimeters are effectively the square against the circle, they will not produce the same intermediate readings as they proceed. Those linear and angular velocities have different implications, and they do not change at the same rates.

The size of the angle the polar planimeter measures depends upon the rate at which it turns about itself as the boundary it is measuring ducks and dives to change the length of the radius at that point. So there will be times when the reading on the polar planimeter—which is the method of tensors—is not incrementing in its area particularly fast because it is not rotating so quickly through the relevant angle; while the linear one—the method of fluxes—keeps busily augmenting its boundary incrementals. This is happening because the polar one is currently busy extending its radius, which does not add—yet—to the sector area. The polar one will, of course, eventually record an increased area. It will do so when it returns to increasing its angle, thus sweeping through a bigger sector. But that has yet to transpire.

By the same token, therefore, measuring the population by counting all the distinct entities, and then determining their average over a given number, which is effectively to determine a radius, need not produce the same results, from moment to moment, as directly measuring the complete population fluxes. The linear planimeter goes its own sweet way while the polar planimeter goes its … but they will end up with the same total value, regardless.

When we measure the population using the polar planimeter and the method of tensors, we measure individual plants and their separate contributions to internal energy until values repeat, no matter how long that might take. We are not so much concerned with the time this takes as with ensuring that there has been the full range of activities that make up a generation. We want to observe the full temporal distribution of biological activities no matter how long each specific activity might take. A polar planimeter's circle is always 360°, and is independent of the boundary length. It depends only on the radius and a sequence of sectors. So if a radius grows outwards, then the circle might be getting bigger, but we can still only ever turn through 360°. The polar planimeter does not care about the boundary length. It only cares about its radius; the sum of those distributions; and the completing of the 360° that is its circulation.

The linear planimeter, by contrast, cares very greatly indeed about its net boundary length, and so about its total fluxes which is the sum and the total of the population's internal energy. It does not, however, care in the slightest about what is happening to any angles, and nor does it care about any radius, or the quality of activity distributions, or the current relative size of any sectors or segments it describes.

In the same way, if the population's average individual mass, or average individual energy, increases, then this effectively extends the radius. It might now take longer to complete a generation, but the generation is still only completed when seeds reappear and values repeat.

We indeed found, in our experiment, that when plant masses and energies increased, the generation length extended. The boundary length—and so the generation—together extended because the radius increased.

What we measured, in our experiment, was that the rate of turning about, through the generations, demonstrably decreased to cause a temporal delay; and that the generation did this whenever either each distinct plant's mass, or its total stock of internal energy, or both, increased. This immediately extended the time the plants took to traverse the 360° for the generation, with the plants turning more slowly about themselves in this direction, and so within the generation. The generation length and time extended when the mass and/or the plant stock of internal energy increased because both the radius grew, and the boundary length increased … which is then the generation time, T.

We found, in our Brassica rapa experiment, that the time for the generation varied between 28 and 44 days, entirely depending—in inverse correlation—upon plant masses and internal energy stocks … and also as according to these expectations determined from these differences in linear and polar planimeters.

If we want to know the rates concerned … if we want to know just how much the measurements made by the linear and the polar planimeters vary—which is also how the flux and the tensor methods of measurement vary… then we simply set the one over the other and express it as a ratio. Nothing could be easier. We will also be measuring variations in plant internal energies.

The ratio between the two values now gives us

M dT  ⁄  ∫dm̅ dn

and

P dT  ⁄  ∫dp̅ dn

to produce the relative differences occurring in mechanical and nonmechanical chemical energy, respectively … but by contrasting the population and the individual methods. These ratios now tell us the extent of the ducking and the diving … of the relative rotation that a flux engages in, and therefore also the degree to which a generation length is changing in its quantity of mechanical and/or nonmechanical chemical energy.

We can now measure the relevant ratios simply by measuring the difference in our linear and our polar planimeters. The overall totals for each of the methods in each case are again the same, but we can now find the relative rates of increase and decrease per each, in internal energy, at any given time. This ratio is, then, the pressure exerted by what they must each do so they can fulfill what they must all do. It is also measurable in the surroundings. It is the pressure exerted by variations.

The great scientist James Clerk Maxwell, who gave us the electromagnetic theory and much else besides, first studied these relative rotatory phenomena. He called this very ratio the ‘twirl’. The more commonly accepted modern name is ‘curl’. Since Maxwell showed that it is a very important physical quantity, it of course has its own sign: ‘∇ x’. This simply means the relative rates of change in internal energy as measured by the ratio or simultaneous difference between a linear and a polar planimeter. Thus the curl in the mass flux is denoted by ∇ x M, that in the energy flux by ∇ x P.

#### The maxim of succession

Our mass flux is expressed as M = nm̅. The curl for that mass flux—i.e. the difference between the linear and polar planimeters—is stating the relative rates of increase in (a) the mechanical chemical energy aspect of internal energy which is , and (b) the number of partitions applied to that internal energy, which is n. It is the number of biological entities at each point in the generation. The curl states the relative contributions each of these makes to the overall flux.

Since the total flux is M, its change in time, or derivative, at any point is dMdt. That dMdt only tells us the total change. It does not tell us how that change comes about. Since M = nm̅, each of n and must be making its distinct contribution to the whole. There is no reason why these should always be the same. The population is very likely changing overall, but each of n and could possibly be making different individual and partial contributions to that overall change. At one point one could be making a greater contribution, and at another point the other could be. The curl is the differences in any changes of n and over time.

The curl in the mass flux measures two things happening in this situation. It measures both the total changes in both mass and number, and their individual changes with respect to each other. Each therefore has its distinct or partial infinitesimal increment—or differential—that is its contribution. These are given different symbols. Where the entire infinitesimal increment at any point has the symbol d, any partial infinitesimal increments contributing to that entire or whole one are each given the symbol ‘∂’. By convention, therefore, we have ∂ for the contribution that mass makes to the overall infinitesimal change, and ∂n for the contribution that numbers make. These are ‘partial differentials’ because they are each a part of an overall change..

Now we know how to separate the distinct contributions that mass and number each make to any overall change, we can compare them directly to each other. We can then determine the relative contributions they make to any ongoing situation. If we express them as rates either proportionately to each other, or else over time, then they are each ‘partial derivatives’. Thus while the overall population change in the mass flux, at any moment, is the derivative dMdt, that overall population change is composed of the two partial derivatives or distinct contributions ∂/∂t and ∂n/∂t.

Since we have defined the curl as the difference between each of the specific or partial rates of change or derivatives over time, then we can now say that ∇ x M = ∂/∂t - ∂n/∂t. This simply says that the rate at which the population mass flux changes, at any time, depends upon the distinct, or individual, rates of change in (a) the individual masses, and (b) numbers in the population. These are all things we can easily measure.

Now … this is most interesting. We have here a clear—but inverse (through that negative sign)—dependency of mass flux on number. That is what the term -∂n/∂t states. That rate of change of numbers is also as trivial to measure as the mass flux itself. We just count biological entities over time, which is exactly what we do to measure the mechanical chemical energy flux and its mass of chemical components.

That little formula for the curl in the mechanical chemical energy or mass flux—∇ x M = ∂/∂t - ∂n/∂t—is telling us that one of the reasons a biological entity will increase in its internal energy and process increased quantities of mechanical chemical energy, as mass, all about itself is because its DNA instructs it to do so, which is +∂/∂t. This of course comes directly from = Mn, where M is the numerator (the number on top in the fraction) which states the direct dependency.

That = Mn expression has a corollary. It has n in the denominator (the number on the bottom). The curl thus tells us that a second reason a biological entity will increase in its internal energy—and so process more mechanical chemical energy, or a larger mass of chemical components about itself—is when the numbers in the surrounding population have decreased. This is -∂n/∂t. That decline in its fellow members also increases the curl, which is its rate of activity in its mechanical chemical energy. And … that claim is exactly what we want to measure.

A mathematical aside

We should now recollect that we have already proven that the divergence (symbolized by ∇ • ) in the mass flux of mechanical chemical components energy—which is simply the average individual mass because ∇ • MM/n = m̅—increases under the same conditions: i.e. both (a) when the flux increases; and (b) when the numbers decrease, with the latter then again being an inverse correlation. Whenever the numbers decrease, the mechanical chemical energy flux density emulates the curl and increases immediately. We now have some very definite number-dependent attributes and variations we can measure.

We can state the curl in the Mendel pressure or mechanical chemical energy and mass flux in words as:

## The rate at which progeny is produced depends upon the rate at which competition occurs.

#### The maxim of apportionment

On the same basis, the curl in the Wallace pressure or nonmechanical chemical energy flux, P (i.e. difference in rate of change of population and individual values), comes from the ratio ∫P dT /∫dp̅ dn. But since the population's nonmechanical chemical energy flux depends upon both its size and its energy density or work rate in that P/M = W, then the curl here has an additional partial derivative and dependency. The total internal energy depends on three things. It depends on (1) the number of biological entities or partitions in biological internal energy, (2) the number of chemical components maintained within each partition, and (3) the modes and methods of molecular vibrations adopted in each partition. The total internal energy value depends upon the energy density or intensity. The curl in nonmechanical chemical energy is therefore given by ∇ x P = ∂/∂t + ∂W/∂t - ∂n/∂t (or else by the equivalent ∇ x P = ∂/∂t - ∂V/∂t - ∂n/∂t).

Thus the internal energy that a biological entity processes about itself can increase (a) because its mechanical chemical energy or mass of components increases; or else (b) because it changes its chemical configuration or energy conformation through its DNA; but also (c) because its numbers decrease.

A mathematical aside

And we have also already proven that the divergence in the energy flux—which is the average individual Wallace pressure in that ∇ • PP/n = p̅—increases both (a) when the flux increases; and (b) when the numbers decrease.

We can state the curl in the Wallace pressure or energy flux in words as:

## The bioactivity of a biological population is subject to increase from an initial value for one or more of three reasons: (a) increases in mass; (b) decreases in competition. All other increases are due to (c) the essential development of the entity or species.

##### Statement | Discussion

And … if creationism and intelligent design are true then neither changes in the curl nor in the divergence, which are changes due to numbers, should ever affect any population. We can express these more formally as the demand that ∇ • n = 0, that ∇ × n = 0, and that ∂n⁄∂t = 0. The divergence, curl, and partial derivatives involving number must all be zero. These claims are all measurable.

We can now explicitly test the ideas that creationism and intelligent design advocate because we can measure all differences both proportionate and absolute, in rates and quantities, and per individual and per the population. We can compare these between any two populations, all with our linear and polar planimeters, and our tensors. All we have to do now is conduct an experiment—such as we did with Brassica rapa—and get some hard data to see if the claim about the lack of influence through numbers is so.

A mathematical aside

More technically, if a population is free from Darwinian competition and evolution then it should be both solenoidal and irrotational: i.e. have no divergence, and no curl. This is again something measurable and predictable we can test. It was the basis of our experiment.

A mathematical aside

There is another very interesting corollary to our discovery of curl. Again by the Helmholtz decomposition theorem of the vector calculus, the curl joins the divergence to be the second of the two properties that uniquely identify any flux. Therefore, if two populations have the same divergence and the same curl, then they are the same. This means that if we measure two generations or subpopulations ostensibly of the same species, then they should show zero relative divergence, and zero relative curl. If creationism and intelligent design are true, then we should have ∇ • n = ∇ × n = ∂n⁄∂t = 0 at all times.

But the mechanical and nonmechanical energies are linked via visible presence, V, or its inverse of the work rate, W. We have the potential to uniquely define every biological population by its values for its average individual fluxes over the entire generation, which is through m̅’ and p̅’. This means that if Darwin's view is correct, we do not need to have ∇ • n = ∇ × n = ∂n⁄∂t = 0 at all times. Numbers may now vary. They simply need to maintain an average of zero to define any species.

However … if numbers do vary, they must do so inversely with average individual mass and energy, and . If numbers increase, then mass and energy must decrease, and conversely. We can now measure all this directly, and state precise values.