The German translation of the "Calcolo delle variazioni" published by Ernesto Pascal in 1897 gives to American mathematicians in convenient form the best book on the calculus of variations that has, to our knowledge, appeared up to the present time. The book consists of only 150 octavo pages and presents concisely the principal facts of the subject. A valuable feature of the work will certainly be found to be the very excellent and apparently complete bibliography given in connection with brief accounts of the development of the calculus of variations. No book with which we are acquainted could be better adapted to controvert the lay opinion that everything in mathematics is exact and beyond dispute. Again and again the author calls attention to errors made by writers in this field. He often calls attention, too, to gaps which remain still to be filled in the theory, and makes the reader sometimes feel that the results which we already have rest on a rather precarious foundation. The chief fault of the book, from our point of view, is that it sacrifices simple and natural discussion to the pursuit of the end so dear to Italian mathematicians, the greatest possible generality. The apparent purpose of the author is to give an account, absolutely rigorous as far as it goes, of the present condition of the science. That such an end is in the calculus of variations especially difficult to attain appears from the fact that the proofs are not always precise and that the author prefers often to tell us that the work given is not rigorous rather than to attempt to make it so.

In no subject is special specialization growing more imperative than in mathematics; in the midst of difficulty and demand the student should hail with delight the valuable services of a work so admirably adapted to purposes of orientation as Professor Pascal's repertorium promises to be. The author's plan, adhered to without deviation, is to present with regard to each theory of modern mathematics the fundamental definitions and notions, the characteristic necessary theorems and formulae, and citations to the principal works of its bibliography. The definitions are clear and unequivocal; the statements of the theorems, always given without demonstration, are concise and unambiguous; and the bibliographical references are sufficiently full to supply the needs of the general mathematical reader.

The little book on Lie's theory which is before us deserves a hearty welcome. For a short time longer there is still no English book on the subject. Let those of us who know a little Italian peruse the present manual. It is all the easier to start upon because there is not room in it for the dignified style and the almost wearisome elaboration of the greater works brought out under Lie's own auspices. Few authors know so well as Sig. Pascal how to present higher mathematics in didactic form. The range of his mathematical learning is moreover cyclopedic. The rate at which one useful and up-to-date Manuale Hoepli from his pen follows another is remarkable. Signs of haste in production, though not entirely absent, are rare. The 'Manuali Hoepli' are books of size for the pocket. Two pages would go on one of an ordinary octavo. The type, which is beautifully clear, is almost extravagantly large. The purchaser for half-a-crown of the present volume might well be forgiven for expecting only a meagre sketch of first principles, and not much of the analysis, perforce abounding in triple suffixes, etc., which is formidable of aspect even on the ample page of Teubner's Lie-Engel. He will be agreeably disappointed. The work is not unambitious. Its aim is "Without lack of rigour and generality to contain in little space all that forms the basis of this advanced part of pure mathematics." The author has at any rate succeeded in making clear in their complete forms the principles and processes of the general theory. Special theories, and in particular the whole subject of contact transformations, are reserved for a promised further volume.

It is certain that through the profound changes which the critical spirit has made in the foundations of the calculus, even a course intended for those for whom mathematics is a means rather than an aim, cannot but use the new results which have been reached . . . it would therefore exhibit a shortsighted view and little esteem for the ability of the future engineer, to believe that it would be sufficient for them, at least if they can, to learn to operate the calculus in about the way in which a workman knows how to operate a machine made by others, and of which he does not know the inner connections.

A discussion is given of the theory of the integraph of Abdank-Abakanowicz with various improvements and modifications which the author has made in order to enable him to solve special types of first and second order differential equations, such as the general linear first order equation, the equation $y^{\prime\prime}=\large(\frac{y^3}{x})\normalsize^{1/2}$, etc. The instruments are more compact but much less flexible than a differential analyser. The input unit for x (a two-wheeled carriage which rolls parallel to the x-axis and which carries the entire instrument), the input and output units for the dependent variables (small carriages carrying pointers and pencils, respectively, which roll on tracks attached to the main carriage), and the integrator (a friction wheel which constrains the motion of the output pencil) are retained in substantially the form used in the original integraph. The modifications consist mainly in the different types of linkages which are used to connect the integrator and the input and output units. These linkages involve curved tracks, and cams, as well as parallelograms.