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,