Supposing we have defined the operation
x + (a − 1),
the operation x + a will be defined by the equality
(1) x + a = [x + (a − 1)] + 1.
We shall know then what x + a is when we know what x + (a − 1) is, and as I have supposed that to start with we knew what x + 1 is, we can define successively and 'by recurrence' the operations x + 2, x + 3, etc.
This definition deserves a moment's attention; it is of a particular nature which already distinguishes it from the purely logical definition; the equality (1) contains an infinity of distinct definitions, each having a meaning only when one knows the preceding.
Properties of Addition.—Associativity.—I say that
a + (b + c) = (a + b) + c.
In fact the theorem is true for c = 1; it is then written