Sec. 3.  Inductive proofs are usually classed as Perfect and Imperfect.  They are said to be perfect when all the instances within the scope of the given proposition have been severally examined, and the proposition has been found true in each case.  But we have seen (chap. xiii.  Sec. 2) that the instances included in universal propositions concerning Causes and Kinds cannot be exhaustively examined:  we do not know all planets, all heat, all liquids, all life, etc.; and we never can, since a man’s life is never long enough.  It is only where the conditions of time, place, etc., are arbitrarily limited that examination can be exhaustive.  Perfect induction might show (say) that every member of the present House of Commons has two Christian names.  Such an argument is sometimes exhibited as a Syllogism in Darapti with a Minor premise in U., which legitimates a Conclusion in A., thus: 

      A.B. to Z have two Christian names;
      A.B. to Z are all the present M.P.’s: 
    .’.  All the present M.P.’s have two Christian names.

But in such an investigation there is no need of logical method to find the major premise; it is mere counting:  and to carry out the syllogism is a hollow formality.  Accordingly, our definition of Induction excludes the kind unfortunately called Perfect, by including in the notion of Induction a reliance on the uniformity of Nature; for this would be superfluous if every instance in question had been severally examined.  Imperfect Induction, then, is what we have to deal with:  the method of showing the credibility of an universal real proposition by an examination of some of the instances it includes, generally a small fraction of them.

Sec. 4.  Imperfect Induction is either Methodical or Immethodical.  Now, Method is procedure upon a principle; and if the method is to be precise and conclusive, the principle must be clear and definite.

There is a Geometrical Method, because the axioms of Geometry are clear and definite, and by their means, with the aid of definitions, laws are deduced of the equality of lines and angles and other relations of position and magnitude in space.  The process of proof is purely Deductive (the axioms and definitions being granted).  Diagrams are used not as facts for observation, but merely to fix our attention in following the general argument; so that it matters little how badly they are drawn, as long as their divergence from the conditions of the proposition to be proved is not distracting.  Even the appeal to “superposition” to prove the equality of magnitudes (as in Euclid I. 4), is not an appeal to observation, but to our judgment of what is implied in the foregoing conditions.  Hence no inference is required from the special case to all similar ones; for they are all proved at once.

There is also, as we have seen, a method of Deductive Logic resting on the Principles of Consistency and the Dictum de omni et nullo.  And we shall find that there is a method of Inductive Logic, resting on the principle of Causation.