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8 been shown to hold good for b = 1, and it may be verified analytically that if it is true for b = &beta; it will be true for b = &beta; + 1. The proposition is thus established by recurrence.

DEFINITION OF MULTIPLICATION.

We shall define multiplication by the equalities: (1) a × 1 = a. (2) a × b= [a × (b − 1)] + a. Both of these include an infinite number of definitions; having defined a × 1, it enables us to define in succession a × 2, a × 3, and so on.

PROPERTIES OF MULTIPLICATION.

Distributive.—I say that (a + b) × c = (a × c) + (b × c). We can verify analytically that the theorem is true for c = 1; then if it is true for c = &gamma;, it will be true for c = &gamma; + 1. The proposition is then proved by recurrence.

Commutative.—(1) I say that a × 1 = 1 × a. The theorem is obvious for a = 1. We can verify analytically that if it is true for a = &alpha;, it will be true for a = &alpha; + 1.

(2) I say that a × b = b × a. The theorem has just been proved for b = 1. We can verify analytically that if it be true for b = &beta; it will be true for b = &beta; + 1.

IV.

This monotonous series of reasonings may now be laid aside; but their very monotony brings vividly to light the process, which is uniform,