If after this cursory survey we review the ground we have covered, we shall recognize the following general relations: In each case the development of a science consists in correlating concepts formed from definite abstractions derived from experience; and by this means we achieve in our minds a mastery over certain parts of our experiences. Such correlations are termed according to the degree of their universality and reliability, rules or laws. A law is the more important, the more definite its statement concerning the greatest possible number of things; and the more accurately it consequently permits of predicting the future. Every law is based upon incomplete induction and is therefore liable to modification by experience. Hence the development of science is of necessity twofold.
In the first place actual relations are examined to see whether or not new relations other than those already known may not be discovered, i. e., constant relations between individual peculiarities. This is the inductive method. And because the possibilities of experience are unlimited it must ever be an incomplete method.
In the second place, relations discovered by induction are applied to cases which have not yet been investigated. Cases resulting from the combination of several inductive laws are particularly liable to be studied. If the combination is correctly made and if the inductive laws are absolutely certain, the result has a claim to unconditional validity. This is the limiting case which all sciences strive to approach. It is almost attained by the simplest sciences, mathematics and certain parts of mechanics. This is termed the deductive method.
In the actual practise of every science both methods of investigation constantly alternate. The best method to discover new and significant inductions is to make a deduction even though its basis be insufficient, requiring subsequent proof from experience. Sometimes the investigator is not conscious of the separate steps of his deduction. In such cases scientific instinct is spoken of. On the other hand, great mathematicians have informed us that they used to find their general laws by induction, by trying and considering individual cases, and that their deductive derivation from other known laws is an independent operation which at times did not follow until much later. Even today there are a number of mathematical propositions which have not reached the second stage and which are therefore at present of a purely inductive and empirical character. The part that such laws play in the sciences rapidly increases as we pass up the series.
Another peculiarity which may be mentioned here is that in the series all preceding sciences assume the characteristics of applied sciences in respect to succeeding ones because they are essential to the course of the last without being themselves increased. They are merely helps to the latter.
