In preparing their _Text-Book of Algebra_[19] the authors, assuming that mental discipline is of the first importance to every student of mathematics, have endeavored to present the elements of the science in a clear and logical form, while yet keeping the needs of beginners constantly in mind. Special attention is given to making clear the reason for every step taken; each principle is first illustrated by particular examples, and then rules and suggestions for performing the operation are laid down. The authors have endeavored to avoid apparent conciseness at the expense of clearness and accuracy, and have thereby made their volume somewhat larger than ordinary text-books. Features to which attention is called are the development of the fundamental operations with algebraic numbers and the concrete illustrations of these operations; the use of type forms in multiplication and division and in factoring; the application of factoring to the solution of equations; the solutions of equations based upon equivalent equations and equivalent systems of equations; the treatment of irrational equations; the discussions of general problems and the interpretation of positive, negative, zero, intermediate, and infinite solutions of problems; the treatment of inequalities and their applications; the outline of a discussion of irrational numbers; a brief introduction to imaginary and complex numbers; and the great number of graded examples and problems. [Footnote 19: Text-Book of Algebra, with Exercises for Secondary Schools and Colleges. By George Egbert Fisher and Isaac J. Schwatt. Part I. Philadelphia: Fisher & Schwatt. Pp. 683. Price, $1.25.] The material of the _Primary Arithmetic_, Number Studies for the Second, Third, and Fourth Grades, of _A. R. Hornbrook_ (American Book Company), has been chosen with careful reference to the development of the number sense of little children, as noticed by the author and as reported by many other observers. A distinctive feature of the work is the use of diagrams called "number tables," as a concrete basis for the child's thinking while he is getting his first ideas of the facts of the addition and multiplication tables. In them the numbers up to one hundred are presented in columns of tens, and so handled as to exhibit to the child's conception the relations of the several digits. By their use he first learns the properties of ten, then of two, and so on of the others--not presented in regular order, but with a view of exhibiting special properties--and their relations to one another. The method is ingenious and appears useful.