(_a_ + _b_) + ([gamma] + 1) = _a_ + (_b_ + [gamma] + 1) = _a_ + [_b_ + ([gamma] + 1)],

which shows, by a series of purely a.n.a.lytic 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._--1 I say that

_a_ + 1 = 1 + _a_.

The theorem is evidently true for _a_ = 1; we can _verify_ by purely a.n.a.lytic 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.

2 I say that

_a_ + _b_ = _b_ + _a_.

The theorem has just been demonstrated for _b_ = 1; it can be verified a.n.a.lytically that if it is true for _b_ = [beta], it will be true for _b_ = [beta] + 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 a.n.a.lytically that the equality is true for _c_ = 1; then that if the theorem is true for _c_ = [gamma], it will be true for _c_ = [gamma] + 1.

The proposition is, therefore, demonstrated by recurrence.

_Commutativity._--1 I say that

_a_ 1 = 1 _a_.

The theorem is evident for _a_ = 1.

We verify a.n.a.lytically 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 proven for _b_ = 1. We could verify a.n.a.lytically that if it is true for _b_ = [beta], it will be true for _b_ = [beta] + 1.

IV

Here I stop this monotonous series of reasonings. But this very monotony has the better brought out the procedure which is uniform and is met again at each step.

This procedure is the demonstration by recurrence. We first establish a theorem for _n_ = 1; then we show that if it is true of _n_ - 1, it is true of _n_, and thence conclude that it is true for all the whole numbers.

We have just seen how it may be used to demonstrate the rules of addition and multiplication, that is to say, the rules of the algebraic calculus; this calculus is an instrument of transformation, which lends itself to many more differing combinations than does the simple syllogism; but it is still an instrument purely a.n.a.lytic, and incapable of teaching us anything new. If mathematics had no other instrument, it would therefore be forthwith arrested in its development; but it has recourse anew to the same procedure, that is, to reasoning by recurrence, and it is able to continue its forward march.

If we look closely, at every step we meet again this mode of reasoning, either in the simple form we have just given it, or under a form more or less modified.

Here then we have the mathematical reasoning _par excellence_, and we must examine it more closely.

V

The essential characteristic of reasoning by recurrence is that it contains, condensed, so to speak, in a single formula, an infinity of syllogisms.

That this may the better be seen, I will state one after another these syllogisms which are, if you will allow me the expression, arranged in 'cascade.'

These are of course hypothetical syllogisms.

The theorem is true of the number 1.

Now, if it is true of 1, it is true of 2.

Therefore it is true of 2.

Now, if it is true of 2, it is true of 3.

Therefore it is true of 3, and so on.

We see that the conclusion of each syllogism serves as minor to the following.

Furthermore the majors of all our syllogisms can be reduced to a single formula.

If the theorem is true of _n_ - 1, so it is of _n_.

We see, then, that in reasoning by recurrence we confine ourselves to stating the minor of the first syllogism, and the general formula which contains as particular cases all the majors.

This never-ending series of syllogisms is thus reduced to a phrase of a few lines.

It is now easy to comprehend why every particular consequence of a theorem can, as I have explained above, be verified by purely a.n.a.lytic procedures.

If instead of showing that our theorem is true of all numbers, we only wish to show it true of the number 6, for example, it will suffice for us to establish the first 5 syllogisms of our cascade; 9 would be necessary if we wished to prove the theorem for the number 10; more would be needed for a larger number; but, however great this number might be, we should always end by reaching it, and the a.n.a.lytic verification would be possible.