The author simply draws out the obvious definition based on highschool physics (density = mass/volume), on Newton (to correct Einstein's erroneous and misleading Assumption of Convenience, that time may be regarded as the fourth dimension), and on a preliminary tightening up of the first three commonly accepted dimensions using a linguist's exacting standards. Then, for good measure (pun), Einstein is invoked to identify the sixth dimension.

Volume I identifies all six of these "dimensions" or senses of measurement, on the basis of linguistic factors, as-yet unconnected highschool physics, the first two definitions in Newton's Principia, and Einstein's famous e = mc². The full, tightened, six-set paradigm (there are more than six but we'll pause here and take time to get everyone "on board"), in the form of a series of stepped formulas, is, first simple, then annotated:

nothing x perceptibility = speck (prerequisite for measurement)
speck x length = line (1st measurement, or "dimension")
line x width = area (2nd dimension)
area x height = volume (3rd dimension, of which space is a special case)

These are the familiar "three dimensions." But notice now that we have "neatened them up." there's a new feature: they now have not only cardinality, but ordinality. Order is now important - and fixed. It is not optional or arbitrary or capricious. No longer can we say, lwh, or hwl, or whl, or hlw, etc. Each builds on the previous and is lethally flawed without it. Or if we were running the series in reverse, each subtracts from the previous and is meaningless without it (for example, volume x *flatth= area, area x *narrowth = line, line x *shortth = speck, speck x invisibility = nothing). So we can build further without "creeping ourselves out" with woo-ooooo, "higher dimensions"! -

volume x density (or velocity"', of which gravity is a special case) = mass (4th dimension, of which matter is a special case)
mass x velocity' = momentum (5th dimension, of which time is a special case)
momentum x velocity" = energy (6th dimension, of which light energy is a special case)
  Note that skipping momentum gives us: mass x velocity' x velocity" = energy, and substituting c (light) for v:  mcc=e or e=mc².

Annotated Presentation     (Format:  dimensional step, then basic formula or equation, then notes)

    to 0: nothing x perception = speck,
where we could also say, nothing x notice = something (as in "I think there's something there")
or we could say, the invisible x perceptibility = the just barely made out (but not enough to measure)

0 to 1:  speck x length = line,
where "speck" (or "point") has zero dimensionality (that is, perceptibility without measurability) and in this basic formula functions as a unit or a one (1), while on a graph it functions as the origin, that is, zero (0),
and where the term "line" is introduced on the analogy of the next two steps, to differentiate the "climbing terms" (such as length, width, and height) from the "plateau terms" (such as line, area, and volume)

1 to 2:  line x width (or breadth) = area (or field or plane)

2 to 3:  area x height (or depth) = volume (or space or room)

3 to 4:  volume x density = mass (or matter or bulk, or 'body at rest'), from d=m/v in every highschool physics book and from Definition I at the beginning of Newton's Principia :
The quantity of matter is the measure of the same, arising from its density and bulk conjunctly.
Note that Newton's reference to bulk as space in his explanatory paragraph clarifies its meaning of "volume."
Let's translate Definition I step by step:
The quantity of matter is the measure of the same, arising from its density and bulk conjunctly.
The quantity of mass is the measure of the same, arising from its density and volume conjunctly.
The quantity of mass is the product arising from its density and volume multiplied together.
Compact:  Mass is density multiplied by volume.
Mathematize:  Mass equals density times volume.
In symbols:  m = d x v
Transpose right-side terms:  m = v x d
Transpose sides:  v x d = m
Desymbolize terms:  volume x density = mass: QED

4 to 5:  mass x velocity (uniform or balanced) = momentum (or 'body in uniform motion'), from M=mv in every highschool physics book and from Definition II at the beginning of Newton's Principia :
The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjunctly.
Let's translate this step by step:
The quantity of momentum is the measure of the same, arising from the velocity and quantity of mass conjunctly.
The quantity of momentum is the product arising from its velocity and mass multiplied together.
Compact:  momentum is velocity multiplied by mass.
Mathematize:  momentum equals velocity times mass.
In symbols:  M = V x m
Transpose right-side terms:  M = V x d
Transpose sides:  V x d = M
Desymbolize terms:  mass x velocity = momentum : QED
The qualification that the velocity here must be balanced or uniform is taken up in Newton's Definition III:
The vis insita, or innate force of matter, is a power of resisting, by which every body, as much as in it lies, endeavours to persevere in its present state, whether it be of rest, or of moving uniformly forward in a right line.
Naturally, there is no way to ascribe rest or uniform motion to a body except in relation to a minimum of five bodies. For example, given only Earth (E) and Moon (M), there is no way to ascribe rest or uniform motion to either, except that if M is at rest, E must be in uniform motion around it, and vice versa. So we can say EM (Earth is circled by Moon) or ME (Moon is circled by Earth) where the "-form" in "uniform" is a smooth circle (more precisely, a smooth ellipse).
However, introduce the Sun (S) and Venus (V), for example, and we have the same ambiguity relative to each and every pair, making for some 12 two-sets;
that is: uniform motion characterizes EM (Earth is circled by Moon) or ME (Moon is circled by Earth); or ES, SE; EV, VE; MS, SM; MV, VM; SV, VS.
This constitutes a "grammar" that accounts for the observed data with one "rule" of 12 parts.
If we ascribe rest to Earth and motion to the other bodies, and introduce non-uniform motion, we can reduce this grammar:
I.e: There is uniform motion between EM, ES.
There is non-uniform motion (ellipse with epicycle) between EV.
This gives us a grammar of two rules: one with two parts and one with one part.
However, if we ascribe rest to the Sun instead of anything else, we can eliminate the non-uniform motion, explain all the data with just uniform motion again, and reduce the original grammar even further:
I.e: There is uniform motion between SE, SV, and EM.
This gives us a grammar of one rule with three parts. Being simpler, it is therefore the better grammar, having captured more powerful generalizations, that is, deeper levels of generalization (or despecification) that still account for all the data - and can generate potential data that we can now look for.
(See Chomsky's Syntactic Structures for this grammar&rule talk and the concept that the best grammar is the one with the fewest rules because it reveals the "most powerful generalizations." Furthermore, we believe that Chomsky's thinking here reveals the mechanism of the repeated extension of self-interest, a vital process in human progress, since "there is no such thing as altruism or selflessness, only specifiably extended self-interest," or in scientific terms, "there is no such thing as objectivity, only specifiably extended subjectivity" or viewpoint.)

Also, M=mv, where M is momentum, m is mass and v is velocity, is a generalization of the first half of Einstein's famous e=mc², rewritten e=mcc, generalized to e=mvv, and split into e=Mv" and M=mv'. Note that c is the velocity of light, and as such, it differs from a generalized dimensional concept by having a specific value: 186,000 mi/sec. So the sixth dimension, energy, at least in Einstein's supposedly relativistic conceptualization, is tied to his absolutization of the fastest human medium of perception, sight (and light).
The best way to regard time is as a plateau concept like line, area, volume, mass, momentum and energy, rather than as a transitional concept like length, width, height, density, velocity and celerity. However, although on the same plateau level as momentum, time is a special case of momentum, and momentum is a general case of time.
Time is a reference momentum similar to the reference mass embodied in the platinum-iridium pound cylinder maintained in London or the platinum-iridium kilogram cylinder maintained at Sèvres; that is, time is a special case of momentum, with its reference mass, surface mark and reference-line termini neglected, specifically (in the small unit, the day), a rotational momentum referenced to the mass of planet Earth as measured by an arbitrary reference mark on its surface, the Greenwich meridian, and a reference line between its center and an arbitrary outside point, the shadowless noonday sun - that is, the reference mass of planet Earth and its rotational velocity judged to be constant enough by the passage of Greenwich directly under the noonday Sun, have been neglected to the point of partial amnesia, but with standardization and repeated refinement based on observation, averaging, and now, particle decay.)

Time, as the all-pervasively, comprehensively quantifying measure, is also the Great Integrator and the Great Equalizer. As the Great Leveller, time answers the same question of all deeds, great or small : "how long did it last?" and it treats them all the same way it treats inactivity : "how long did it last?" It treats the lives of great people the same as ordinary people - "how many years did they live?" It treats the lives of people the same as insects - "how long do they live?"

In our opinion Einstein mistreated the concept of time dimension. Our argument goes like this.
"Relative" and "absolute" are opposites. "Relativity" and "absoluteness" are opposites. But Einstein betrayed his Theory of Relativity when he absolutized the speed of light, regardless of viewpoint, or speed and direction of viewpoint. Basically this was a type of anthropomorphism or anthropocentrism (in psychologeze, "projection") because it absolutized our best (fastest), commonly agreed, human perceptual sense, namely, sense of sight.
Einstein's fundamental metaphor, riding on a light beam, got him off to a bad start because it mixed together and merged subject and object, observer and observed, and confused all subsequent thinking. It introduced an unaccountability, as when a child asks 'why?' and a parent says, 'because I say so.'
To absolutize, or constantize, the energy dimension in terms of light energy, he then had to move the flexibility somewhere else, and he wound up sacrificing the constancy (in terms of the unidirectionality and irreversibility) of the time dimension and thereafter coming up with all kinds of nonsense, like arriving back from a light-speed trip before leaving, etc.

5 to 6:  momentum x celerity (non-uniform, unbalanced velocity) = energy.
Note also that the shortcut span from dimension 4 to 6 is given by the general form of Einstein's famous equation,
e = mc² or mc²=e or mv²=e.
Note also that a skewed traverse of the span from dimension 4 to 6 is given by:
mass x acceleration = force, which may be regarded as a 1-1/2 level climb from 4 to 5-1/2.
force x displacement = work (i.e: vectored energy), which may be regarded as a 1/2 level climb from 5-1/2 to 6.
If Newton had completed the three-set, his third definition, instead of changing the subject, would have been something like the following: The quantity of energy is the measure of the same, arising from the celerity and quantity of motion conjunctly.

Note that this extended version of the dimensional paradigm accountably relates, indeed interweaves, the four modern "elements" of space (3rd), time (5th), matter (4th) and energy (6th). (Compare the four ancient elements of air, earth, fire and water.)

In supporting Newton's identification of mass rather than Einstein's convenience of time as the fourth dimension, and identifying time, non-arbitrarily, as a special case of the general fifth-dimensional variable, momentum, Volume I is potentially the most sensational of the three volumes. The book then explores some light that the now-established standard dimensional hierarchy of the six basic senses of measurement alias "dimensions" can cast on human progress, and its potential coincidence with social evolution as divided into six great social-science ages (discussed in Vol.II).

On the fourth dimension, see also websites -

• *http://www.geocities.com/CapeCanaveral/7997/ (3/21/2003)
  Eric Saltsman's (esalts@zoomnet.net) "The fourth dimension" (A: hypercube or tesseract, which he seems to regard as synonyms, and which we regard merely as one way of diagramming density or mass. The diagram on the left on http://www.geocities.com/CapeCanaveral/7997/hypercube.html is the best form of the hypercube to illustrate density, which at its most basic, is simply "filling," or something inside something else)
  
• *http://www.mninter.net/~jmcmanus/4d/
  J. McManus' "On the fourth dimension"
• *http://www.hypermaths.org/quadibloc/math/fdiint.htm
  John Savard's "The fourth dimension" page
• *http://users.powernet.co.uk/bearsoft/GnRel.html
  Bruce Harvey's Alternative Physics site at least has gone as far as "My second objection to the General Theory of Relativity would be that time is simply not the fourth dimension."
• *http://www.virtualchaos.org/science/intro2tp.html
  marvellous example of the impressive paragraphs you can produce without establishing an accountable paradigm

On the sixth dimension, see also websites -
[we haven't even begun on this one, so again, let us know at timesizing@aol.com if you find some good websites.]