The active power is always the substitution of equals, and it is an accident that in a pair of equalities we can make the substitution in two ways. From *a* = *b* = *c* we can infer *a* = *c*, either by substituting in *a* = *b* the value of *b* as given in *b* = *c*, or else by substituting in *b* = *c* the value of *b* as given in *a* = *b*. In *a* = *b* ~ *d* we can make but the one substitution of *a* for *b*. In *e* ~ *f* ~ *g* we can make no substitution and get no inference. In mathematics the relations in which terms may stand to each other are far more varied than in pure logic, yet our principle of substitution always holds true. We may say in the most general manner that *In whatever relation one quantity stands to another, it stands in the same relation to the equal of that other.* In this axiom we sum up a number of axioms which have been stated in more or less detail by algebraists. Thus, “If equal quantities be added to equal quantities, the sums will be equal.” To explain this, let *a* = *b*, *c* = *d*. Now *a* + *c*, whatever it means, must be identical with itself, so that *a* + *c* = *a* + *c*. In one side of this equation substitute for the quantities their equivalents, and we have the axiom proved *a* + *c* = *b* + *d*. The similar axiom concerning subtraction is equally evident, for whatever *a* - *c* may mean it is equal to *a* - *c*, and therefore by substitution to *b* - *d*. Again, “if equal quantities be multiplied by the same or equal quantities, the products will be equal,” For evidently *ac* = *ac*, and if for *c* in one side we substitute its equal *d*, we have *ac* = *ad*, 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 Q*a* denote in the most general manner that we do something with the quantity *a*; then if *a* = *b* it follows that Q*a* = Q*b*.