The theorem has just been demonstrated for b = 1; it can be verified analytically that if it is true for b = β, it will be true for b = β + 1.
The proposition is therefore established by recurrence.
Definition of Multiplication.—We shall define multiplication by the equalities.
(1) a × 1 = a.
(2) a × b = [a × (b − 1)] + a.
Like equality (1), equality (2) contains an infinity of definitions; having defined a × 1, it enables us to define successively: a × 2, a × 3, etc.
Properties of Multiplication.—Distributivity.—I say that
(a + b) × c = (a × c) + (b × c).
We verify analytically that the equality is true for c = 1; then that if the theorem is true for c = γ, it will be true for c = γ + 1.
The proposition is, therefore, demonstrated by recurrence.
Commutativity.—1º I say that
a × 1 = 1 × a.
The theorem is evident for a = 1.
We verify analytically that if it is true for a = α, it will be true for a = α + 1.
2º I say that
a × b = b × a.
The theorem has just been proven for b = 1. We could verify analytically that if it is