tion unless it had a root. And since the mathematician made the equation, and called it such, it had a root if the mathematician said that it had. To discuss the question was thus as useless as to discuss the existence of the six-winged fairy in the fairyland of your own creation. My friend would only admit the significance of inquiring what the value of any of the n roots actually in question might be.
And, as a fact, of course, my friend’s argument, despite its quarrel with the labors of Gauss and the other algebraists, had its own relative force. A theory of algebraic quantities is conceivable which should arbitrarily begin one of its sections by defining certain symbols as the roots of algebraic equations, and which should then proceed to demonstrate the properties of these symbols, as well as of the equations in terms of the symbols. Such a method of procedure has indeed been proposed as a formal device in the course of the more recent history of the theory of equations. But as an historical fact, the mathematicians, in the first place, actually proceeded otherwise, defined, apart from the general theory of equations, their realm of algebraic quantities, both of those called “real” and of those called “complex,” defined also their general equations, and then, indeed, had upon their hands the problem of proving that within that realm of the algebraic quantities, as thus previously defined, there could be found such as would furnish their general equation with roots. Hereupon, indeed, the resulting problem was one whose solution was no longer, like the creation of the six-winged fairy, a matter of arbitrary choice. The
