formu Now the way which exact science has hit upon historically to Mathesecure this unification is to analyze what happens into smaller matical units so selected as to be common constituents in a vast variety lation and complexity of events, and constituents moreover of such a the ansort that these more complex happenings in all their variety swer to and difference can be regarded as groupings of these common this need units after a quantitative pattern. This means that in any field in which such an analysis is successfully made, every verified law of connection between gross qualitative substances or events becomes viewed as a specific consequence of a few more general laws expressing the behavior of these common units. Any complex event being thus regarded as a quantitative combination of the simpler units, all exact laws become reducible to mathematical form, gaining thus the objectivity and precision of a quantitative statement in symbolic language, as well as a common foundation with all other laws applying in the same field. It is easy to see that the conceptions of cause and effect become almost obsolete wherever science has advanced to the point of successfully picking out these quantitative units in any given process of change. The events that had previously been called cause and effect, in other words, become conceived now as themselves in constant change, and the universal relation between them becomes an expression, under special conditions, of the more general laws of change that are constantly revealed in the endless rearrangements of their internal parts. The reason is thus evident for the statement made earlier that dependence on causal explanations is the mark of a relatively early stage in the development of a science, the stage at which this profounder analysis has not yet been successfully made. ysis and common It is further important to observe that with the introduction Relation of this method of quantitative analysis on a broad scale, we ap- between pear for the first time to introduce a noticeable gap between such analthe assumptions of common-sense verification and those of scientific procedure, and it is the sense of this gap that in large part explains what popular distrust of science still exists. By its re- planations duction of our sense world into a complex of minute, invisible units, physical science seems to replace the realm observed by sense ex Summary RIGHT THINKING the common man by a quite different, unfamiliar, even mysterious world. The statement that the sun warms, as verified by common experience, gives nobody trouble; in confirming it we know precisely what facts of life the terms used refer to, and our thinking is implicitly controlled by the same principles as the scientist appears to be using in the examples of his work so far cited. But if we substitute for this simple statement one in terms of the correlation between the play of certain ether waves upon the surface of the body and an increased rapidity of the motion of the molecules of the latter, many people feel themselves transported into another realm which has quite left behind the familiar facts with which we began. The reader will see, however, that this gap is apparent rather than real. Even common sense analyzes in its explanations, picking out facts to be related that would otherwise remain unnoticed, as the experiment with the shoe indicates. A thoroughgoing quantitative analysis of the experienced world but carries forward this analysis in a more minute and systematic way, and gives it the form that is most fruitful for exact understanding and control. But it is obvious that if this is the manner in which we are to seek the necessary unity of science, we must so apply our observation and experiments as to take for granted this quantitative reduction, and lead to the formulation in quantitative terms of the laws of connection which result. The gradual realization of this need has led to the adoption, where possible, of a more inclusive and satisfactory technique, in which all other scientific procedures merge, and in which they find, so far as we can judge at present, their culmination. This we shall term the method of functional analysis. BIBLIOGRAPHY DE KRUIF, P., Microbe Hunters. A historico-biographical description of the great discoveries in the field of bacteriology, as thrillingly written as a novel. DESCOUR, L., Pasteur and His Work. The briefest treatment of Pasteur's work available in English which affords an adequate portrayal of his scientific method. FOWLER, T., Inductive Logic. An old but exceedingly serviceable analysis of the processes of inductive generalization. MILL, J. S., A System of Logic, part III. The classic formulation of the canons of inductive logic. ROBINSON, D. S., Illustrations of the Methods of Reasoning, chap. V. A group of brief illustrations of inductive generalization. The term function Vague and mathematical functions CHAPTER ELEVEN THE METHOD OF FUNCTIONAL ANALYSIS WHAT is a function? In the sense in which science uses the term, and in which we may appropriately speak of a functional explanation, it has two meanings. For one thing, it means the normal or characteristic activity of any distinguishable entity, as when we speak of the function of a lever, of the lungs, of a public official. With this meaning we are not especially concerned in inductive logic. The other meaning, with which we are directly concerned, must also be distinguished into a broader and a more limited connotation, the common idea in both of them being that of a regular relation between two or more processes of change. Let us examine these meanings rather carefully. In the broader sense, a function may be defined as any fact so related to another that it varies in some determinate way with that other. Either may then be called a function of the other, which means that for every distinguishable change in the one there corresponds a distinguishable change in the other. Thus we may say that the general health of a person is a function of his digestion, that the crops of a given region are a function of its soil and climate, or that the duties of Congress are a function of the Constitution. But it is evident that such statements carry little exactitude, and that they no more help to bring about the unification of scientific knowledge than the methods of formulation previously noted. To achieve these results, as has just been shown, we must analyze each of the related processes into quantitative units, and express the relation between them as a relation of variable magnitudes. This gives us mathematical precision in our state ments of scientific law, and also offers a basis for uniting in a single system of exact knowledge all processes which are analyzable in terms of the same type of unit. The laws of behavior of these units thus become the general laws pervading the entire field so united. In other words, we turn from this broader conception of function to a more limited but far more fruitful conception, that of mathematical function. It is functional explanations of this sort that are increasingly seen to mark the maturity of exact science. First we must define a few basic terms. A mathematical func- A few tion is a variable magnitude so related to another magnitude definithat for every value of the latter there is a corresponding value tions of the former. By value, of course, is meant any determinate quantity of such a magnitude. Either magnitude may be spoken of as a function of the other, but since in any given statement of their relation one must be arbitrarily regarded as primary and the other spoken of as its function, it is necessary to draw a distinction between independent and dependent variables. The independent variable is that magnitude which in the statement in question is regarded as given, or is arbitrarily assigned. The other magnitude or magnitudes, which are regarded as functions of it, are dependent variables. Thus we may say that the surface and volume of a sphere are functions of its radius. This means that for any value of the radius, say nine inches, there correspond definite values for the surface and volume, respectively, expressed in square or cubic inches. When stated in this form the radius. is the independent variable, since it is taken arbitrarily as given, and the surface and volume, being affirmed to be functions of the radius, are dependent variables. It would be just as correct to convert the law and say that the radius is a function of the surface or of the volume. In this case the radius would be a dependent variable, and whichever of the other two were taken as given would become the independent variable. In such relations as these there is only one independent variable, since a determinate value for it is all that is needed to determine completely the values of the others. Sometimes there must be more than one independent variable in order that def |