39. The Law of Continuity.
From the fact that natural phenomena in general proceed continuously we can deduce a number of important and generally applicable conclusions which are constantly used for the development of science.
When a relation of two continuously varying values of the form A = f(B) is conjectured, we convince ourselves of its truth by observing for different values of A the corresponding values of B, or reversely. If we find that changes in the one correspond to changes in the other, the existence of such a relation is proved, at first only for the observed values, though we never hesitate to conclude that for the values of A lying between the observed values, but themselves not yet observed, the corresponding values of B will also lie between the observed values. For example, if the temperature at a given place has been observed at intervals of two hours, we assume without hesitancy that in the hours between when no observations were made, the values lie between the observed values. If we indicate the time in the usual manner by horizontal lines and the temperature for the general periods of time by longitudinal lines, the law of continuity asserts that all these temperature points lie in a steady line, so that when a number of points lying sufficiently near one another is known, the points between can be derived from the steady line which may be drawn through the known points. This very commonly applied process will yield the more accurate results the nearer the known points are to one another, and the simpler the line.
The application of the law of continuity or steadiness, therefore, means no less than that it is possible, from a finite, frequently not even a very large, number of individual results, to obtain the means of predicting the result for an infinitely large number of unexamined cases. The instrument derived from this law, therefore, is an eminently scientific one.
The value of this instrument is still greater if it succeeds in expressing the relation A = f(B) in strict mathematical form. First, the result of the determination of a number of individual values of that function is represented as a table of co-ordinated values. By the graphic process above described, or by its equivalent, the mathematical process of interpolation, this table is so extended that it also supplies all the intermediate values. But this is still a case of a mechanical co-ordination of the corresponding values. Often we succeed, especially in the relation of simple or pure concepts, in finding a general mathematical rule by which the magnitude A can be derived from the magnitude B, and reversely. This is the only instance in which we speak of a natural law in the quantitative sense.
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.