An introductory text-book of logic - Mellone, Sydney - Rare Books at ...

THE GENERAL NATURE OF INDUCTION. 249
ing, which is a fairly simple example, is given by De
Morgan :
"
Observe the proof that the square of any number is
equal to as many consecutive odd numbers, beginning
with unity, as there are units
6x6=1 + 3 + 5 + 7 + 9 + 11.
in that number : thus
Take any number, ,
and write down n X n dots in rank and file, so that a dot
represents a unit. To enlarge this figure into (+ i) X
(ti + i ) dots, we must place n more dots at each of two
adjacent sides, and one more at the corner. So that the
square of n is changed into the square of (+i) by
adding 2/2+1, which is the (+i)th odd number.
(Thus 100 Xi oo is turned into 101 X 101 by adding
the joist odd number, or 201). If then the alleged
theorem be true of n X it is therefore true of , (+ i) X
(+ But it is true of the first
i). number, for i x i = i
;
therefore it is true of the second i.e., 2x2 1 + 3; and
and so on."
therefore of the third i.e., 3X3 = 1 + 3 + 5;
Here we have a series of terms (i, 2, 3, &c.) in which
we know the relation between every pair of consecutive
terms. We wish to establish a fact about every term in
it. We suppose that the fact holds of any one term,
which we therefore denote by n ; and prove that it holds
of the next term, which is +i. We then find by
observation that it holds of the first term, i ; therefore
it must hold of the second, 2 ;
sality of the result depends
and so on. The univer
on the fact that the essential
relation (which is simply a numerical one) between any
pair of consecutive terms is known, as , n + 1 ; and the
proof depends on this alone.
On the other hand, where this proof from the essential
conditions cannot be obtained, we may verify a theorem
in case after case, without being sure that it holds
universally. This is a case of
"
incomplete induction