The Method of Residues gives three distinct results: First, it tells what is left over after all the other parts of the phenomenon have been explained. Second, it tells how much is left over, and third, it calls attention to the unexplained parts of the phenomenon. For example, in the first concrete illustration, by subtracting the known quantities from the total quantity, what is left over is found to be coal; not only so but we are able to calculate the exact amount of coal. This illustrates the first and second results of the Method of Residues. (Like concomitant variations it is seen that residues is serviceable in given definite quantitative values.) The discovery of Neptune illustrates well the third result of this method; i. e., after accounting for every other force, it was found that there was yet a force at work which had never been explained. It is this third feature of unexplained residues which has placed “Science in its present advanced state.” “Most of the phenomena which nature presents are complicated; and when the effects of all known causes are estimated with exactness, and subducted, the residual facts are constantly appearing in the form of phenomena altogether new, and leading to the most important conclusions.” So says John Herschel. Almost all of the discoveries in astronomy have come about in this way. If a heavenly body does not behave as it should according to the established theory, then either thetheory is wrong or there is some residual phenomenon which needs to be explained. Its suggestiveness is, therefore, the most important function of this method, though this very feature is the one which makes evident its greatest disadvantage. The unexplained residual phenomenon may be very complex and, therefore, a careless observer is apt to overlook a lurking element which in reality is the true cause.
7. THE GENERAL PURPOSE AND UNITY OF THE FIVE METHODS.
Thinking has been defined as the deliberative process of affirming and denying connections. It is obvious that these five methods are a matter of affirming and denying connections between antecedents and consequents. As soon as the looked for connections are established, the antecedents and consequents are known to be related to each other as causes and effects. In this attempt to find and prove connections the Method of Agreement is chiefly valuable in suggesting workable hypotheses, and the method of difference in verifying, through experiment, the correctness or incorrectness of these hypotheses.
In substance the principle conditioning both methods is this: “If a single antecedent is invariably present when the phenomenon is present and invariably absent when the phenomenon is absent then this antecedent is the cause of the phenomenon.” To put it still more briefly: Between two phenomena there is a causal connection, if the conjunction between the two is invariable. It isthe business of Agreement to single out the one antecedent and of Difference to show, by presenting the negative as well as the affirmative side of the case, that the conjunction of the one antecedent and the particular phenomenon is invariable. The Joint Method is merely a combination of Agreement and Difference carried into more varied and complex situations. The methods of Concomitant Variations and Residues are merely modifications of Difference; the former being used when the chief feature is the fluctuation of the phenomenon, and the latter when it is desired to find what is left over.
Agreement suggests the hypothesis, “difference” proves it; the joint method is “difference” more or less complicated, concomitant variations is “difference” applied to fluctuating phenomena, residues is “difference” used to find what and how much is left over.
Agreement is the method of observation and belongs to the physician and nature student. Difference and the Joint Method are experimental devices which are used by the physicist and chemist. Concomitant Variations is the method of unstable phenomena and naturally attaches itself to the economist and statistician. Residues is the method of “lurking exceptions” and is favored by the astronomer and mathematician.Residues, being the method of “what is left over,” is the most common in daily affairs.[14]
All the five methods are forms of inductive thinking which lead to the establishment of causal connections bymeans of the principle of the invariable conjunction of phenomena.
8. OUTLINE.
THE FIVE SPECIAL METHODS OF OBSERVATION AND EXPERIMENT.
(1) Aim of Five Methods.