NATURE of all ages will have been aroused ; he will, no doubt, have drawn from the portraits in the book very vivid and characteristic conclusions as to the personal appearance of men like Klein, Fermat, and Gauss; and, being a lucky American boy, his interest will have carried him yet further to the biographies in the school library. If this is indeed his gain, it is no small acquisition. In most cases it is to be hoped that he will have learned to dispense with the irritating "Why? " which

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... pers the pages with the doubtful stimulus of a confession of weakness. (3) Prof. Barnard's volume is' very straightforward and is clearly written. The difficulties of beginners are not unknown to him, and in the selection and construction of examples he claims to have borne in mind the advantage of extensive numerical applications and the necessity of constant appeal to fundamental principles. From the outset he brings in the use of limits "as the only satisfactory way of defining such quantities .as velocity and acceleration." The advantages of vector analysis are very much in the (Australian) :air at the moment, so we have a chapter on the "merest beginnings " of the elements, which is to be regarded as giving " an altern?-tive method of dealing with questions connected with parallelogram liiws." The value of what may be called a merely incidental reference to the use of a powerful tool may be questioned. "0 the little more :and how much it is ! " The direct treatment of simpJe harmonic motion as rectilinear motion under a given law of force is justified by the statement that "it has the advantage that the student is not led to imagine that some special circle has to be thought of in connection with the motion, as is so commonly the case in the common method." About one-fifth of the book is devoted to rigid dynamics. Altogether it is a very interesting endeavour to smooth the path of the beginner who is to continue his study of the subject when provided with the additional weapon .afforded by the calculus. W. J. G.