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6 which was trodden slowly by the founders of the science.

Why is so long a preparation necessary to habituate oneself to this perfect rigour, which it would seem should naturally be imposed on all minds? This is a logical and psychological problem which is well worthy of study. But we shall not dwell on it; it is foreign to our subject. All I wish to insist on is, that we shall fail in our purpose unless we reconstruct the proofs of the elementary theorems, and give them, not the rough form in which they are left so as not to weary the beginner, but the form which will satisfy the skilled geometer.

DEFINITION OF ADDITION.

I assume that the operation x + 1 has been defined; it consists in adding the number 1 to a given number x. Whatever may be said of this definition, it does not enter into the subsequent reasoning.

We now have to define the operation x + a, which consists in adding the number a to any given number x. Suppose that 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 what x + a is when we know what x + (a - 1) is, and as I have assumed that to start with we know what x + 1 is, we can define successively and "by recurrence" the operations x + 2, x + 3, etc. This definition deserves a moment's