# Page:Popular Science Monthly Volume 17.djvu/538

But I wish to call attention to the fact that here we find the best, the most satisfactory introduction to the study of modern algebras, modern mathematics. When told that in these systems a product may not vary with each of its factors; that a product may vanish without either of its factors vanishing; that subtraction and division may be indefinite; that, in fact, any system, e. g., quaternions, where the products and powers of the units are themselves linear functions of the units, excludes the ordinary assumption that a product shall vary with each of its factors; that from q q, ${\displaystyle =}$ o, it does not follow that either q ${\displaystyle =}$ o or q, ${\displaystyle =}$ o; that a quadratic equation, e. g., in quaternions, besides its sixteen roots proper, may have an indefinite number of roots which arise from the fact that the process of division is not a definite one; when told these, and very many more such, the beginner is only too sure to think, "This is a hard saying," and may give up the subject in hopeless confusion. If, however, he will start with Schroeder, "Der Operationskreis des Logikkalkuls," he will find the clearest explanation and illustration of these things contained in his own every-day thoughts about the commonest objects; and, while learning an elegant logic, will be mastering, perhaps, the most exquisite dual algebra.