(1) Principle stated.
As stated by Mill the principle of residue is this: “Subtract from any phenomenon such part as is known by previous inductions to be the effect of certain antecedents, and the residue of the phenomenon is the effect of the remaining antecedents.”
In simpler form the notion is this: Subtract from any phenomenon those parts of it which are known to be the effect of certain antecedents, and what is left of the phenomenon is the effect of the remaining antecedents.
(2) Principle symbolized.
| Antecedent | Consequent |
| A | x |
| B | y |
| C | z |
The total cause of the phenomenon xyz is ABC. But it is known that the cause of x is the antecedent A; whereas the cause of y is the antecedent B; hence it is concluded that the cause of z is the antecedent C.
(3) Concrete illustrations.
Problem: To find the weight of coal.
| Antecedents | Consequents | ||||||
| Weight of driver, | = | 4200 lbs. | |||||
| Weight of wagon, | |||||||
| Weight of coal. | |||||||
| Weight of driver, | = | 200 lbs. | = 2200 lbs. | ||||
| Weight of wagon. | 2000 lbs. | ||||||
Hence we may conclude that the weight of coal is 4200 lbs. − 2200 lbs., or 2000 lbs.
Perhaps the most noted instance in history of the application of this method, was the one which resulted in the discovery of Neptune. In calculating the orbit of Uranus, it was found that the combined attractions of the sun and the known planets did not account for the path which Uranus took. There was some unknown influence at work. Assuming that this unaccountable attraction was due to the presence of another planet beyond the orbit of Uranus, an Englishman by the name of Adams, and later the Frenchman Leverrier, were able to indicate by the principle of Residues, the spot where this planet should be. By directing the telescope toward this point, Neptune was discovered.
(4) Distinguishing features:
The phenomenon always occurs,
The antecedents are usually invariable,
Some of the antecedents are known to be the cause of a part of the phenomenon.
(5) Advantages and disadvantages.
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 the theory 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.
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 is the 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 by means of the principle of the invariable conjunction of phenomena.
THE FIVE SPECIAL METHODS OF OBSERVATION AND EXPERIMENT.
(1) Aim of Five Methods.
Fundamental fact of causation.
Aim of analysis.
Methods of
agreement
difference
joint
concomitant variations
residues
(2) Method of Agreement.
Principle stated
Method symbolized
Method illustrated
Distinguishing features of method
A matter of observation and experiment
Advantages and disadvantages
(3) Method of Difference.
Principle stated
Method symbolized
Method illustrated
Advantages and disadvantages
Characteristic features
Agreement and Difference compared
(4) The Joint Method of Agreement and Difference
Principle stated
Method symbolized
Concrete illustrations
Distinguishing features
Advantages and disadvantages
(5) Method of Concomitant Variations
Principle stated
Method symbolized
Concrete illustrations
Distinguishing features
Advantages and disadvantages
(6) The Method of Residues
Principle stated
Method symbolized
Concrete illustrations
Distinguishing features
Advantages and disadvantages
(7) General Purpose and Unity of Five Methods
One fundamental principle
(1) The fundamental fact of causation underlies the three forms of induction, but is most conspicuous in the method of analysis and may be ascertained by recourse to one of the experimental methods.
(2) The principle of the method of agreement may be summed up in the two statements: The sole invariable antecedent of a phenomenon is probably its cause and the sole invariable consequent of a phenomenon is probably its effect. These two statements may be symbolized and illustrated.
The essential characteristics of the method of agreement are the phenomenon always occurs; there is at least one invariable antecedent; the other antecedents vary.
The method of agreement together with the other four methods may justly be termed methods of experiment as well as methods of observation.
The difficulties of the method of agreement are in the main plurality of causes, immaterial antecedents, complexity of phenomenon and uncertainty of conclusion. These difficulties may be summarized as involving a phenomenon which may have several causes; may be preceded by conditions of no causal consequence; may be so involved as to prevent exhaustive examination; and may give unreliable conclusions.
Agreement is valuable chiefly in furnishing to the investigator plausible hypotheses.
(3) The principle of difference is this: “Whatever is invariably present when the phenomenon occurs and invariably absent when the phenomenon does not occur, other circumstances remaining the same, is probably the cause or the effect of the phenomenon.”
Like agreement, difference admits of symbolization and illustration by concrete examples.
The chief difficulties attending difference are: in nature varying one antecedent at a time is infrequent, and it is easy to overlook antecedents which are closely related to the case under investigation.
Difference is the most common method of the experimental sciences. The characteristic features of difference are, the phenomenon does not always occur, one antecedent is variable, while the others are invariable.
The methods of agreement and difference are complementary processes. Agreement attempts to eliminate all the antecedents but one, while difference aims to eliminate one only. Agreement is a method of observation, while difference is a method of experiment. The conclusion of the method of difference gives greater certainty than that of the method of agreement.
(4) The joint method may be stated in this way: Among many instances if one circumstance is invariably present when the phenomenon occurs and invariably absent when the phenomenon does not occur, this circumstance is probably the cause or the effect of the phenomenon.
The instances of the joint method are more numerous and more varied than those of either agreement or difference.
The joint method has the distinguishing characteristics of both agreement and difference.
Because it furnishes greater opportunities for multiplying and varying the instances involved, the joint method presents fewer objections than either of the two separate methods.
The positive branch of the joint method suggests the hypothesis, while the negative branch proves it. This makes the method somewhat ideal.
(5) The principle of concomitant variations may be stated as follows: If when one phenomenon varies alone, and another also varies alone, the one is either the cause or the effect of the other. This is the method of fluctuation, and is used when it is impossible to make the phenomenon disappear altogether, as in the case of difference.
The chief function of concomitant variations is to establish exact quantitative relations between cause and effect.
(6) The principle of residues is this: Subtract from any phenomenon those parts of it which are known to be the effect of certain antecedents, and what is left of the phenomenon is the effect of the remaining antecedent.
The most valuable feature of residues is its suggestiveness; an attempt to explain the “residual phenomenon” has led to many important scientific discoveries.
(7) The five methods are concerned with the establishment of causal connections between phenomena. Agreement suggests the connection while difference proves it. The other methods are modified applications of difference, necessitated by some peculiar form which the phenomenon may take. A statement of the one principle involved is: “If the conjunction between two phenomena is invariable then there is a causal connection.”
All of the methods are forms of inductive thinking.
(1) Explain “the fundamental fact of causation.”
(2) Show that the fact of causation is most conspicuous in induction by analysis.
(3) Name the five special inductive methods of observation and experiment.
(4) State, symbolize, and illustrate the method of agreement.
(5) Give examples of antecedents which do not function as causes.
(6) Show that the “special methods” are a matter of both observation and experiment.
(7) Give the distinguishing features of the method of agreement; illustrate by reference to the symbols.
(8) Exemplify the plurality of causes; immaterial antecedents; complexity of phenomenon.
(9) Show that the conclusions of agreement are largely hypothetical.
(10) State, symbolize, and illustrate the method of difference.
(11) Show by illustration that, in the method of difference, only one antecedent should be varied at a time.
(12) Show that difference is naturally a method of experiment.
(13) Explain Bacon’s use of the term “crucial instances.”
(14) Name and explain the characteristic features of the method of difference.
(15) Show that agreement and difference are complementary.
(16) Explain and illustrate the joint method.
(17) What inference may be drawn from the following instances:
| Antecedents | Consequents |
| A L M T | p q r g |
| B L M E | z q r x |
| B C M E | r z x y |
| A M T H | p q g o |
(18) “Mr. Darwin, in his experiment on cross and self fertilization in the vegetable kingdom, placed a net about one hundred flower heads, thus protecting them from the bees. He at the same time placed one hundred other flower heads of the same variety of plant where they would be exposed to the bees. He obtained the following result: The protected flowers failed to yield a single seed. The others yielded about 2,720 seeds. Thus cross-fertilization was proved.” (Hibben).
What method did Darwin employ? Symbolize the experiment.
(19) Summarize the distinguishing marks of the joint method.
(20) Show that the joint method is more ideal than either agreement or difference.
(21) State and give concrete illustrations of the law of concomitant variations.
(22) What is the chief function of concomitant variations? Illustrate.
(23) Give instances where it would be impossible to use difference, but easy to use concomitant variations.
(24) Explain this: “The quantitative variation between antecedent and consequent may be either direct or inverse.”
(25) State and explain by illustration the method of residues.
(26) What are the advantages and disadvantages of the principle of residues?
(27) State the principle which virtually sums up the five methods.
(28) Write briefly on the practical applications of the five methods to the ordinary walks of life.
(1) Trace the connection between the method of agreement and induction by simple enumeration.
(2) Show that Mill’s methods may properly be termed “Inductive Methods of Scientific Investigation.”
(3) How may it be shown by “agreement” that the high cost of living is due to the tendency to spend more than we earn?
(4) Assume that you are a member of the Board of Health, and that you desire to ascertain the cause of the diphtheria epidemic by means of the principle of agreement.
(5) What is the error involved in coming to the conclusion that to sit at table where there are thirteen, may mean the death of one of the thirteen before the end of the year.
(6) Indicate how it could be shown, by the method of difference, that the mosquito is responsible for the propagation of yellow fever.
(7) “Another experiment similar to this was tried by Plateau, who put some food of which cockroaches are fond on a table and surrounded it with a low circular wall of cardboard. He then put some cockroaches on the table; they evidently scented the food, and made straight for it. He then removed their antennae.” (Hibben). Complete and give with explanations the method used.
(8) “In some cases it is impossible to remove an element which is supposed to be the cause of an effect under investigation.” Explain and illustrate.
(9) “Extreme care must be taken that, in the withdrawing of any element, no other element is inadvertently introduced.” Tyndale supposed he had proved spontaneous generation, when, after sealing in a jar of boiled water a wisp of baked hay, he found, after many days, indications of life within the bottle. In transferring the hay to the bottle, he carried the former across the room. What element was inadvertently introduced?
(10) “The attempt to determine the numerical relations according to which two phenomena vary, requires the utmost caution as soon as our inference outsteps the limits of our observations.” (Fowler). Explain this in connection with the law of concomitant variations.
(11) “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.” Make clear by illustration this quotation which has reference to the principle of residues.
(12) Explain “invariable conjunction of phenomena.”
(13) Investigate by means of one of the five methods the following problems:
(1) “All vegetables which grow to root should be planted during the last two days of the waxing moon.”
(2) “In this section the south wind is the storm wind.”
(3) “Mischief is the outcome of misdirected energy.”
(4) “Bad boys usually receive unjust treatment.”
(5) “An ounce of prevention is worth a pound of cure.”