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effected. One equation is active and the other is passive, and it is a pure accident of this form of inference that either equation may be indifferently chosen as the active one. Precisely the same result happens in this case to be obtained by a similar act c is the active equation,

=

of reasoning in which b

as shown below:

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a = b

C

a.

My warrant for this view of the matter is to be found in the fact that the negative form of the axiom is now easily brought into complete harmony with the affirmative form, except that, since it has only one equation to work by, there can be only one active equation and one form in which the inference can be exhibited as below:

α

S hence S

C

C.

--

Inference is seen to take place in exactly the same manner as before by the substitution of a for b, and the negative equation or difference bc is the part in which substitution takes place, but which has itself no substitutive power. Accord

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ingly we shall in vain throw two differences into the same form, as in

as b

S or

a =

bsc

S

a,

because we have no copula allowing us to make any substitution.

17. I am confirmed in this view by observing that, while the instrument of substitution is always an equation, the forms of relation in which a substitution may be made are by no means restricted to relations of equality or difference. If a = b, then in whatever way a third quantity c is related to one of them, in the same way it must be related to the other. If we take the sign to denote any conceivable kind of relation between one quantity and another, then the widest possible expression of a process of mathematical inference is shown in the form

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If in one case we take the sign as denoting that c is a multiple of b, it follows that it is a multiple of a; if it is the nth multiple of one, it is the nth multiple of the other; if it is the nth submultiple,

or the nth power, or the nth root of one, it similarly follows that it stands in the same relation to the other; or if, lastly, c be greater than b by n or less than c by n, it will also be greater or less than a by n. In this all-powerful form we actually seem to have brought together the whole of the processes by which equations are solved, viz. equal addition or subtraction, multiplication or division, involution or evolution, performed upon both sides of the equation at the same time. That most familiar process in mathematical reasoning, of substituting one member of an equation for the other, appears to be the type of all reasoning, and we may fitly name this all-important process the substitution of equals.

A

18. An apparent exception to the statement that all mathematical reasoning proceeds by equations may perhaps occur to the reader, in the fact that reasoning can be conducted by inequalities. chapter on the subject of inequalities may even be found in most elementary works on algebra, and it is self-evident that a greater of a greater is a greater, and what is less than a less is less. Thus we certainly seem to have in the two formula, a>b>c hence a> c,

and

a <b<c hence a <c,

two valid modes of reasoning otherwise than by

equations. But it is apparent, in the first place, that the use of these signs < and > demands some precautions which do not attach to the copula = ; the formulæ,

a > b < c,
a <b>c,

do not establish any relation between a and c; and I think the reader will not find it easy to explain why these do not and the former do, without implying the use of an equation or identity. The truth is, that the formulæ,

a>b>c,

a<b<c,

involve not only two differences, but also one identity in the direction of those differences, whereas the formulæ,

a>b<c,

a<b>c,

appear to fail in giving any inference because they involve only differences both of direction and quantity.

Strength is added to this view of the matter by observing that all reasoning by inequalities can be represented with equal or superior clearness and precision in the form of equalities, while the con

trary is by no means always true. inequality

Thus the

a> b

is represented by the equality

(1) a = b+p,

in which is any positive quantity greater than zero; and the inequality

b> c

is similarly represented by the equality (2) b = c + 9,

in which q is again a positive quantity greater than zero. By substituting for bin (1) its value as given in (2), we obtain the equation

a = c + p + 9,

which, owing to the like signs of p and q, is a representation in a more exact and clear manner of the conclusion

a > c.

On the other hand, the formula

a>b<e

would evidently lead to the equation

a = e + p−r,

the

in which is the excess of a over b, and excess of e over b. Now this equation, taken in

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