Induction is a matter of universalizing less universal experiences. In this the process may assume any one of three forms, namely: (1) Induction by simple enumeration; (_inductio per enumerationem_); (2) Induction by analogy; (3) Induction by analysis.
Having observed a few instances the generalization is, “All birds have wings.” The certitude of this may now be strengthened by observing more birds and finding without exception that each has wings.
By noting on Mars geometric markings which resemble canals, the generalization is vouchsafed that Mars is inhabited by human beings. Other similarities in atmospheric conditions, existence of land and water, etc., tend to make this generalization more plausible.
By analyzing water taken from a certain spring, it is found to contain hydrogen and oxygen in the proportion of 1 to 8; in consequence a generalization to this effect is posited. Analyses of specimens from other sources yield similar results and thus the generalization is given greater certitude.
As a usual thing the particular form which the induction assumes depends on the nature of the topic under investigation and also on the mental make-up of the investigator. The general statement that all birds have wings could hardly be derived by means of analogy or analysis, but is a matter of a casual observation of many instances. Moreover, that mind given to accurate observation, but not inclined to note resemblances or to carry on experiments, would naturally follow the first inductive type. On the other hand, simple enumeration would be impossible in questions like the habitability of Mars, and would yield no results in cases requiring definite scientific experimentation like electrolysis.
It is worthy of note that some topics lend themselves to all three modes of procedure. To wit: (1) Enumeration. Without being taught the rule the child is given a list of examples involving the dividing of a decimal by a decimal and is asked to solve them. By comparing his answers with those in the book, he somewhat accidentally discovers what seems to be the correct rule for pointing off in the quotient. By following this rule and each time comparing answers he establishes the truth. (2) Analogy. If .24 ÷ .6 is the first example, the child may resort to the well known process of dividing a common fraction by a ( 24 60 24 4 ) common fraction, ( ――― ÷ ――― = ―― = ――, ) then, because of their close ( 100 100 60 10 ) resemblance, he may reason that decimal fractions should yield the same result. (3) Analysis. Here the child reasons that since division of decimals is the inverse of multiplication of decimals, the rule for pointing off might be the inverse of the multiplication rule. By trying this out and proving his answer in each example, he becomes convinced of the correctness of his reasoning.