Calcul des Probabilites

This book by M. Levy on the Theory of Probability is written in order to emphasise a point of view which seems to have been ignored by other writers on the subject. I ts special feature is the prominent place given to the use of the characteristic function which Cauchy was the first to introduce. M. Levy at tempts to justify the fundamental principles of the theory of errors by giving a suitable precision to the intuitive idea of accidental error, and by deducing in a rigorous manner tha t the accidental error obeys the law of Gauss. Several writers think tha t this result does not justify the mathematical apparatus necessary to establish it, and the author at tempts to combat this idea by showing that the mathematical methods required are not so formidable as is generally supposed. The book is divided into two parts, the first of which is elementary and is devoted to setting forth the principles of the theory of probability. Here the author has been careful to state what is definition and what is mathematical reasoning. He has therefore emphasised the part which is subjective in the idea of probability, and he discusses the transition from the subjective to the objective. The second part of the book gives a systematic account of the use of the characteristic function. The conditions, which are necessary in order tha t a function may be a characteristic function, are deduced, and the properties of a given law of probability are obtained from its corresponding characteristic function. The general results obtained are applied to the laws due to Gauss and Cauchy. I t is doubtful whether this portion of the book will be appreciated by readers who have not acquired a considerable knowledge of analytical processes. As an application of the theory of errors M. Levy has devoted a chapter to the Kinetic Theory of gases. Here he gives prominence to the idea of reversibility, from which he deduces Maxwell's law. On the whole M. Levy's work will be considered by many to be rather prolix. Readers who find the first par t new and exciting are likely to be overwhelmed by the analysis in the second part, and those who can follow the second part with ease will find the first part rather dull. But the book contains a fund of valuable information, and the author has certainly succeeded in showing that the use of the characteristic function has a synthetic value which deserves a prominent place in the development of the subject.