This new series is rather pompously announced as striving to build "up the New Humanism on the basis of the student's immediate economic interest and environment" (which implies a considerable modification of the accepted meaning of Humanism). We translate this as meaning that it is intended to give the reader some idea of the various sciences and arts as they find application in industry and commerce. This is a worthy object, and, on the whole, the books are simple and interesting expositions of the utilitarian aspect of the sciences in question. There is, perhaps inevitably, a tendency to hurry over fundamental difficulties which will not, we think, leave an intelligent student satisfied. For instance, to say that a force is whatever changes motion, without further explanation, may well puzzle the reader, who knows that he can push against a heavy stone without producing any apparent motion. However, there is a distinct place for books of this general character, which do good work by showing to a wide audience the peaceful achievements of science and its practical aspects; they act as a counterblast to the deadening tradition of rule of thumb. The industrial chemistry is particularly comprehensive, and has an excellent set of original diagrams of industrial plant. The series is well printed and well illustrated, and, for present times, moderately priced. It deserves wide recognition. PROJECTIVE VECTOR ALGEBRA. By L. SILBERSTEIN. G. Bell & Sons. 7_s._ 6_d._ net. It is not often that a book appears describing an essential advance in pure mathematics which is intelligible to the man of moderate attainments in that science--by moderate attainments we mean such knowledge of mathematics as is picked up in one or two years at a university. Dr. Silberstein, who is well known in this country for his original work, especially in connection with the theory of relativity, has, in his _Protective Vector Algebra_, developed his latest researches in geometry in a form which is attractive and free from pedantic formalities, and has throughout aimed at simplicity of expression, in contrast to certain modern mathematicians who endeavour to lend importance to minor conventional problems by a bewildering display of definitions and theorems. The essential novelty of the book, from which the whole theme is developed, is the generalised definition of the addition of vectors, which does not need any construction of parallel lines, but depends solely on a straight line construction making use of the points where the vectors cut an arbitrary fixed straight line. Dr. Silberstein's definition is a generalisation of the Euclidean one, to which it reduces if the arbitrary line just mentioned is moved away to infinity, and if the space is Euclidean. The knowledge of geometry which is demanded is little more than the usual postulates of projective geometry and Desargues' theorem. From his definition the