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A Syllabus of Plane Algebraical Geometry. By C. L. Dodgson, M.A., Student and Mathematical Lecturer of Christ Church, Oxford.—Oxford: Parker.

The object of this syllabus is to present a sketch of a concise, logical, and complete system of algebraical geometry. Looked at from this point of view this little book is, in our opinion, marked by merits of no ordinary kind. It begins by stating with great precision those definitions, postulates, axioms, and elementary propositions which are required for the application of algebra to geometry. Now everyone who has carefully thought on this subject must have observed how deficient ordinary text-books are in the statement of definitions, and in demonstrations which are applicable to all instances to which they profess to extend. For example, a book which has been so much praised as Salmon's ‘Conic Sections’ attempts no satisfactory explation of the positive and negative signs of the radius vector, nor of those of α and p in the well-known equation, x cos α + y sin a - p = 0. Indeed, in order to do this effectually it would be necessary carefully to extend and modify the ordinary definitions; and although this has been attempted by various authors, including even so eminent a mathematician as Professor De Morgan (see his ‘Differential Calculus’), yet we have seen no book in which this object has been so completely attained as in the book we are now reviewing.

The syllabus does not aim at novelty in its results; but this kind of novelty is often of far less importance than novelty of method. We have heard that one of our most eminent mathematicians said, that the most difficult mathematical book to write would be a logical and philosophical work on arithmetic; and in a similar way, the completeness of the system developed in Mr. Dodgson's book, and his strictly scientific method, places it in our opinion far higher than those numerous text-books like Salmon's ‘Conic Sections,’ whose principal merit consists in importing into our Universities some of the more recent discoveries of continental mathematicians. As this book however contains very few demonstrations, and the reader is referred for them elsewhere, we are afraid that it will only be generally read by those who attend the author's lectures at Christ Church; and we therefore hope Mr. Dodgson will be induced to write a complete work on Analytical Geometry, founded upon the able sketch which has just been published. We think that that sketch bears the same relation to the present state of analytical geometry as an admirable syllabus written many years ago by the late Dean of Ely bore to the then state of trigonometrical science.