If any one expects to find in this book a text-book on algebra like, except in name, to most text-books on that subject he will be disappointed. The book is not a beginners book; it is elementary only in so far as it begins at the beginning. Starting with the theory of proportion as stated by Euclid, the author builds upon this the algebra of real quantities and establishes the laws of combination of such quantities by simple geometrical constructions. After devoting a chapter to the definition and discussion of the circular and hyperbolic functions he takes up the algebra of complex quantities. By means of Argands mode of representation he shows that the laws which were established for real apply as well to complex quantities. At the end of this chapter he states briefly and clearly the characteristics of a logically complete algebra, and incidentally points out that an algebra which admits evolution and the logarithmic process, but precludes the imaginary and the complex quantity is logically only the fraction of an algebra. Then follow three chapters devoted respectively to generalized circular and hyperbolic functions, to graphical transformations and to the properties of polynomials. The first two of these, though interesting in themselves, are digressions from the main argument and might perhaps be omitted without serious injury to the book. The third, however, is more important, for it contains a proof of the so-called fundamental theorem of algebra, viz: that every algebraic equation has a root, a theorem which in most text-books is not proved and in many is totally ignored. In his preface Professor Stringham calls attention to the fact that algebra, unlike geometry, which is a model of exact reasoning, has become a collection of processes practically exemplified and of principles inadequately explained. He has endeavored in his book to do just the reverse and to give to his readers the theory and not the practice of algebra. In our opinion he has succeeded exceedingly well. The first four chapters give a complete and well-reasoned account of the fundamental principles of algebra. Moreover, the book is interesting from the fact that it contains things not found in the ordinary text-books. Such for example are: Euclids theory of proportion, Napiers definition of logarithms, the authors own extension of this definition to complex quantities, and a very complete graphical representation of the analogy between circular and hyperbolic functions. We think it safe to say that both the teacher and the student of mathematics will find the book eminently pleasing and stimulating. --The School Review, Volume 2