and a second similar substitution gives us
ac = bd.
We might prove a like axiom concerning division in an exactly similar manner. I might even extend the list of axioms and say that “Equal powers of equal numbers are equal.” For certainly, whatever a × a × a may mean, it is equal to a × a × a; hence by our usual substitution it is equal to b × b × b. The same will be true of roots of numbers and c√a = d√b provided that the roots are so taken that the root of a shall really be related to a as the root of b is to b. The ambiguity of meaning of an operation thus fails in any way to shake the universality of the principle. We may go further and assert that, not only the above common relations, but all other known or conceivable mathematical relations obey the same principle. Let Qa denote in the most general manner that we do something with the quantity a; then if a = b it follows that
Qa = Qb.
The reader will also remember that one of the most frequent operations in mathematical reasoning is to substitute for a quantity its equal, as known either by assumed, natural, or self-evident conditions. Whenever a quantity appears twice over in a problem, we may apply what we learn of its relations in one place to its relations in the other. All reasoning in mathematics, as in other branches of science, thus involves the principle of treating equals equally, or similars similarly. In whatever way we employ quantitative reasoning in the remaining parts of this work, we never can desert the simple principle on which we first set out.
Reasoning by Inequalities.
I have stated that all the processes of mathematical reasoning may be deduced from the principle of substitution. Exceptions to this assertion may seem to exist in the use of inequalities. The greater of a greater is undoubtedly a greater, and what is less than a less is certainly less. Snowdon is higher than the Wrekin, and Ben Nevis than Snowdon; therefore Ben Nevis is higher than the Wrekin. But a little consideration discloses sufficient reason for believing that even in such cases, where equality does not apparently enter, the force of the reasoning entirely depends upon underlying and implied equalities.
In the first place, two statements of mere difference do not give any ground of inference. We learn nothing concerning the comparative heights of St. Paul’s and Westminster Abbey from the assertions that they both differ in height from St. Peter’s at Rome. We need something more than inequality; we require one identity in addition, namely the identity in direction of the two differences. Thus we cannot employ inequalities in the simple way in which we do equalities, and, when we try to express what other conditions are requisite, we find ourselves lapsing into the use of equalities or identities.
In the second place, every argument by inequalities may be represented in the form of equalities. We express that a is greater than b by the equation
a = b + p, (1)