Text-book of Algebra

Abstract

ON the whole this is a sound and instructive book. In the chapters on first principles the distinction between signs of operation and signs of quality has been very properly emphasised by a special notation, instead of being ignored; the treatment of systems of equations is excellent; and that of surds is much better than usual, although exception might be taken to some of the notation, and the existence of √2 as a definite number cannot be proved (as the authors seem to think) by considering the diagonal of a unit square. The book is rather unequally written, and errors sometimes occur which contrast curiously with the accuracy which generally prevails Thus in the proof of the remainder theorem the same symbol Q is used for two entirely different things; it is assumed without proof that if r is a proper fraction rⁿbecomes infinitesimal as n increases indefinitely; and it should have been stated explicitly that i is a definite symbol obeying the law i² = - I, together with the usual laws of operation, and that if a is positive is understood to mean i. If these last precautions are not taken, it cannot be proved, for instance, that , and, in fact, the authors' treatment of this identity is defective. Then such problems as “factor a + b” are perfectly unmeaning, especially after chapters on surds and complex numbers; probably the answer intended is , but any number of others might be constructed, for instance , and so on. It ought to be unnecessary to say that all questions on factors should be put in a perfectly definite way.

Text-book of Algebra.

By G. E. Fisher I. J. Schwatt Part I. Pp. xiv + 684 (Philadelphia: Fisher and Schwatt, 1898.)