Especially since so much has been treated generally so far, with a sort of overview perspective, it may be useful to note why it is now necessary to start delving into specifics, and in particular why this approach of following through examples with annotations is a good way to handle such details. It has already been emphasized that the main objective is to convey the methods of physics, which are more important than any particular answers they have made possible. That is why modern physics is such an ideal environment in which to pursue this problem, because it makes the methods both visible and indispensible, even though there may still be fewer everyday applications of particular examples from Relativity than there would be of, say, those from elementary Newtonian mechanics. In that sense the importance of the method has been emphasized over the immediate applicability of particular examples.
From such a point of view, it is tempting to assert that the understanding of the method is more important even than all of its results, but this is probably an inappropriate division, because of the degree to which the practice of Physics is still something of an art. This is true in the sense that, while each of the physical laws explaining the behavior of some part of nature is a concrete and useable tool, we have achieved no similar depth of understanding of how people think, and consequently no analogous theory explaining Physics itself. So while we recognize the continuity of method, clearly enough that we are sure that there is something special and powerful about Physics in particular, as much of our understanding is contained in the appreciation of the specific examples as can be contained in any general statements we make to refer to them all.
Yet in viewing the details, it is still more important to see how the method gives them continuity, than to remember any particular conclusions that result. In other words, while progress in physics has often been made through intuited leaps by remarkable people, the structure of the knowledge and reasoning still forms a continuous chain from the elementary all the way to the most powerful ideas. Too often, exposure to the content of modern physics, or even somewhat older but still profound theories, takes on the aspect of a walk through a museum, and in some sense misses the most important realizable benefit of physics, which is that it provides the tools and organization to improve the efficacy and independence with which every person thinks. It is important to understand the degree to which physical knowledge
To see this, one should start at the beginning in discussing the formation of physical theories. By starting at the beginning rather than jumping to the end, we do not compromise our opportunity to present the final answers in their most elegant states of coherence, nor by including their proper context to we diminish the stature of the remarkable people who have been able to to create such ideas. If anything, we can appreciate both more fully, because we are made aware of how much of the deep consistency of nature is available to be noticed in everyday experiences, and yet what insight has been required to extract that continuity in forms we can use as matter-of-factly as we use any other tools.
Starting at the beginning does not mean that we should discuss the Special Theory of Relativity by noting that there were problems with Maxwell's electromagnetic theory, or even with Newton's laws if one tries to reconcile them with the experimental observation that the speed of light is always the same, no matter how it was created or how it was observed. The logical roots of physics, and the epistomological framework which all physicists use, goes much deeper and further back than that, because to understand how to use what he calculates, a physicist must be able to place it within the context of the consequences it generates in experience.
One tool is used more than all of the others combined to put
physical ideas in context. Every physicist, irrespective of his
area of specialization, uses it more than any specialized body
of theory, and it becomes so much a matter of habit that as an
explicit notion it can tend almost to be overlooked. This tool
is called Dimensional Analysis,
and it is the most basic
level at which physics emerges as a distinct way of dealing with
nature, among all the others we use. In this sense, Dimensional
Analysis is properly considered the beginning of physics.
Two things should be noted about Dimensional Analysis as we proceed to describe it, one being its importance as a point of departure and the other its implications as a habitual change in a person's way of thinking. Throughout their existence, people have recognized patterns in the occurrances of nature. It seems likely even that animals do this, though in different and probably less specific ways. Yet a discontinuous change in the power people have to use their ability to recognize patterns occurred in a short period some three hundred years ago, and has persisted since. This change in the usefulness of people's ability to think coincides with what we refer to here as `the emergence of physics'. It does not come from a new intrinsic ability to notice the patterns in nature, because the intrinsic ability has always been present, so it must come from something new about the ways in which people can organize and state those perceptions, to make them concrete and testable from one time to the next and from one person to another, and from the resulting change in their ability to percieve patterns by being attuned to their useful features.
The connection between the publication of Newton's Principia Mathematica and this change in power is widely acknowledged. No such similar `invention' of Dimensional Analysis can be marked with a time, just as no particular author can be solely credited with its original formulation as a concept. Yet the rudiments of Dimensional Analysis as a practice were present in the work of Galilleo that preceded Newton's, and in the later work of Kepler that accompanied it. It may be that the ideas themselves are very old, going as far back as the appreciation of the concept of measurement itself, and that they were suddenly given usefulness by Newton's laws, where they could be seen as the most approximate descriptions of natural systems, those that set the overall scales within which Newton's laws filled in the detailed description. Therefore it seems historically appropriate, we well as logically necessary, that we present Dimensional Analysis before we can even discuss the work of Newton, which has been widely regarded as `the beginning of physics'. Yet at the same time, the reader should understand that Dimensional Analysis is not in itself a well-defined concept or set of rules. Its importance and usefulness can be recognized immediately, but its use also requires a certain amount of judgement which will be greatly developed by the more concrete ideas like Newton's laws which follow, and which is enhanced by practice and experience. So while it is the proper idea to regard as the beginning, it is also, like most physical theories themselves, a fuzzy sort of beginning, which yields great power but also requires refinement in what follows.
At the same time that the incompleteness of Dimensional Analysis as a set of ideas has been acknowledged, it is important to understand how increadibly successfully, as a practice, it is used. This owes to the sense in which it is more fundamental than any of the later and more refined methods we will discuss. This power of simplicity comes from the fact that the world is a complicated place. When we are looking for patterns, by far the most difficult part of the search is extracting from the overwhelming body of available experience the part that is relevant to answering a particular question. This requires sifting through a large range of observations and filtering out the bulk of them in some efficient way. In the first stages of this exercise, most of the detail provided by Newton's laws, or indeed any precise set of laws, is superfluous. What is needed is a general way to make `in-the-ballpark' estimates of how big, how often, how fast, or how many of something we should expect to consider. Dimensional Analysis is the tool that is used to make such estimates of the properties of physical systems. It can be used with great success once one knows something about the underlying laws themselves, which suggest concretely which features to consider. But it can be used with a remarkable level of success even without this intellectual framework, from nothing more than the experience gained through a little practice.
Physicists use it in both ways. When they know which laws are likely to be relevant, they use Dimensional Analysis to generate approximate answers and predictions, very quickly and efficiently. This is tremendously helpful in deciding where to look for the consequences of a particular phenomenon, and for locating mistakes and oversights. This is the primary relevance of Dimensional Analysis to improving the sensitivity of a person to patterns, above whatever `intrinsic' level may have been present without the incorporation of the methods of physics. When they don't know which laws are relevant, Physicists use Dimensional Analysis to uncover the scales characteristic of the patterns they observe, as the first stage toward guessing and testing, and thereby trying to discover the more precise laws.
Both these practices tend to instill in Physicists the habit of noticing, quantitatively and analytically, what goes on around them. As we go further to discuss the particular laws of physics, and especially as we consider their applications in the problems, it will become evident, and hopefully breathtaking, what an increadible range of natural phenomena have been made accessible to the human mind through amazingly few general physical principles. All of the power of that intellectual framework can be harnessed to make the world a knowable place, if one establishes the ties to common (and then also to less common or downright uncommon) experience. Those ties are provided by Dimensional Analysis, and it is their constant presence in the perceptions of the Physicist that probably constitutes the most important difference of his thoughts from the thought processes that were available to anyone before the creation of physics.
That said, we proceed to describe what Dimensional Analysis is, how it is used, and what can be understood about it as a concept in its own right.