and most worthy of our intellectual and æsthetic impulses--these and other propositions are ably and interestingly presented and constitute an essential portion of this very stimulating and clarifying volume. The success of the work is attested by the appearance of this second edition; the chief addition consists of a discussion of the quantitative method as applied to biological phenomena, which the readers of others of the author’s works will recognize as one of his favorite subjects of investigation. _THE TEACHING OF ELEMENTARY MATHEMATICS._ The book with the above title, by David Eugene Smith, principal of the State Normal School, at Brockport, N. Y., contains much of value, presented in a very readable and attractive manner. The subjects treated are arithmetic, algebra and geometry. About half the book is devoted to the first. The author sketches the history of the teaching of arithmetic from the earliest times, gives a critical examination of the different systems which have been tried and aims to discover the correct general principles upon which the instruction should proceed. He notices the tendency of many of our schools to follow too closely the Grube method, or a modification of it. The chapter on the present teaching of arithmetic is full of valuable suggestions. Algebra and geometry are treated in the same way. Much useless lumber is cleared away, and the whole discussion is marked by strong common-sense, an element not always present in discussions of this kind. The extreme differentiation in the teaching of these three branches which prevails in so many schools is condemned. It is urged that the blending of algebraic method and notation with the higher parts of arithmetic, and the early introduction of the inductive study of geometric form, both contribute to the substantial progress and development of the student. Valuable references are given to other writings for fuller discussions on special topics. These references cover works in English, French, German and Italian. _GEOLOGY._