Thus, for example, we can observe what volume a given quantity of air occupies when successively subjected to different pressures. If we arrange all these values together in a table, we can also calculate the volume for all the intermediate pressures. But on close inspection of the corresponding numbers of pressure and volume we notice that they are in inverse ratio, or that when multiplied by one another their products will be the same. If we denote the space by v and the pressure by p, this fact assumes the mathematical form p. v = K, in which K is a definite number depending upon the quantity of air, the unit of pressure, etc., but remaining unchanged in an experimental series in which these factors stay the same. The general functional equation A = f(B) becomes the definite p = K/v. And this formula enables us to determine by a simple calculation the volume for any degree of pressure, provided the value of K has been once ascertained by experiment.
At first we have a right to such a calculation only within the province in which the experiments have been made, and the simple mathematical expression of the natural law has for the time being no further significance than that of a specially convenient rule for interpolation. But such a form immediately evokes a question which demands an experimental answer. How far can the form be extended? That there must be a limit is to be directly inferred from the consideration of the formula itself. For if we let p = 0, then v = infinity, both of which lie beyond the field of possible experience.
Similar considerations obtain in all such mathematically formulated natural laws, and each time, therefore, we must ask what the range of validity of such an expression is, and answer the question by experiment.
While in this discussion the mathematically formulated natural law seems to have the nature only of a convenient formula of interpolation, we are nevertheless in the habit of regarding the discovery of such a formula as a great intellectual accomplishment, which so impresses us that we frequently call it by the name of the discoverer. Now, wherein lies the more significant value of such formulations?
It lies in the fact that simple formulas are discovered only when the conceptual analysis of the phenomenon has advanced far enough. The very simplicity of the formula shows that the concept formation which is at the basis of it is especially serviceable. In Ptolemy's theory of the motion of the planets the means for calculating their positions in advance was given just as in the theory of Copernicus. But Ptolemy's theory was based on the assumption that the earth stands still, and that the sun and the other planets move. The assumption that the sun stands still and that the earth and the other planets move greatly facilitates the calculation of the position of the planets. In this lay the primary value of the advance made by Copernicus. It was not until much later that it was found that a number of other actual relations could be represented much more fittingly by means of the same hypothesis, and thus the Copernican theory has come to be generally recognized and applied.
The significance of the law of continuity and its field of application have by no means been exhausted by what has been said above. But later we shall have a number of occasions to point out its application in special instances, and so cause its use to become a steady mental habit with the beginner in scientific research.
40. Time and Space.
Time and space are two very general concepts, though without doubt not elementary concepts. For besides the elementary concept of continuity which both contain, time has the further character of being one-seried or one-dimensional, of not admitting of the possibility of return to a past point of time (absence of double points) and of absolute onesidedness, that is, of the fundamental difference between before and after. This last quality is the very one not found in the space concept, which is in every sense symmetrical. On the other hand, owing to the three dimensions it has a threefold manifoldness.
That despite this radical distinction in the properties of space and time all of our experiences can be expressed or represented within the concepts of space and time, is very clear proof that experience is much more limited than the formal manifoldness of the conceivable. In this sense space and time can be conceived as natural laws which may be applied to all our experiences. Here at the same time the subjective-human element of the natural law becomes very clear.
The properties of time are of so simple and obvious a nature that there is no special science of time. What we need to know about it appears as part of physics, especially of mechanics. Nevertheless time plays an essential rôle in phoronomy, a subject which we shall consider presently. In phoronomy, however, time appears only in its simplest form as a one-seried continuous manifoldness.