But we shall not take it up; it is foreign to our purpose; all I wish to insist on is that, not to fail of our purpose, we must recast the demonstrations of the most elementary theorems and give them, not the crude form in which they are left, so as not to harass beginners, but the form that will satisfy a skilled geometer. DEFINITION OF ADDITION.--I suppose already defined the operation _x_ + 1, which consists in adding the number 1 to a given number _x_. This definition, whatever it be, does not enter into our subsequent reasoning. We now have to define the operation _x_ + _a_, which consists in adding the number _a_ to a given number _x_. 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 _a_ + (_b_ + 1) = (_a_ + _b_) + 1, which, apart from the difference of notation, is nothing but the equality (1), by which I have just defined addition. Supposing the theorem true for _c_ = [gamma], I say it will be true for _c_ = [gamma] + 1. In fact, supposing (_a_ + _b_) + [gamma] = _a_ + (_b_ + [gamma]), it follows that [(_a_ + _b_) + [gamma]] + 1 = [_a_ + (_b_ + [gamma])] + 1 or by definition (1) (_a_ + _b_) + ([gamma] + 1) = _a_ + (_b_ + [gamma] + 1) = _a_ + [_b_ + ([gamma] + 1)], which shows, by a series of purely analytic deductions, that the theorem is true for [gamma] + 1. Being true for _c_ = 1, we thus see successively that so it is for _c_ = 2, for _c_ = 3, etc. _Commutativity._--1º I say that _a_ + 1 = 1 + _a_. The theorem is evidently true for _a_ = 1; we can _verify_ by purely analytic reasoning that if it is true for _a_ = [gamma] it will be true for _a_ = [gamma] + 1; for then ([gamma] + 1) + 1 = (1 + [gamma]) + 1 = 1 + ([gamma] + 1); now it is true for _a_ = 1, therefore it will be true for _a_ = 2, for _a_ = 3, etc., which is expressed by saying that the enunciated proposition is demonstrated by recurrence. 2º I say that _a_ + _b_ = _b_ + _a_. The theorem has just been demonstrated for _b_ = 1; it can be verified analytically that if it is true for _b_ = [beta], it will be true for _b_ = [beta] + 1. The proposition is therefore established by recurrence.