The author simply draws out the obvious definition based on Newton (to correct Einstein's convenient but erroneous assignment of time to the fourth dimension), highschool physics and a preliminary tightening up of the commonly accepted three dimensions using a linguist's exacting standards. Then, for good measure (no 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 largely linguistic factors, such as the first two definitions in Newton's Principia and Einstein's famous mc²=e. They seem to have been missed because they are so glaringly obvious but cloaked in the archaic English of Andrew Motte's early translation of the original Latin of the Principia. The full, tightened, six-set paradigm (there are more than six but we're pausing here), in the form of a series of stepped formulas, is, first simple, then annotated:

dot x length = line (first dimension or measurement)
line x width = area (second dimension)
area x height = volume (third dimension)
volume x density = mass (fourth dimension, of which weight is a special case)
mass x velocity' = momentum (fifth dimension, of which time is a special case)
momentum x velocity" = energy (sixth dimension)
 

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

0 to 1:  dot x length = line,
where "dot" (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 functioning 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 (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.
(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." For our part, we assert that Chomsky's process explains the mechanism of extending self-interest, a vital function for 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.)

5 to 6:  momentum x celerity = energy. Note that celerity is non-uniform or unbalanced velocity.
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.

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 with 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, Vol. 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.]