The appearance of this Review is a significant indication of the enormous development of mathematical studies in recent years. Nearly every one who has attempted to keep himself abreast of mathematical research has been obliged sooner or later to recognise the practical impossibility of mastering the literature of every branch, and has resigned himself to a comparatively elementary study of the general subject while devoting his main energies to special departments. Even then, so numerous are the Societies that publish Proceedings and Transactions, and so varied are the Journals that are chiefly mathematical in their content, that it is no easy matter for the mathematician to get a knowledge of what is being done in any special field by workers outside (sometimes in) his own country. The need for a publication that will, without undue delay, furnish a conspectus of the literature of the subject is thus a very real one. That the need has been felt is sufficiently shown by the synopses of the contents of other Journals, given in such publications as Darboux's Bulletin, and more especially by the excellent Jahrbuch über die Fortschritte der Mathematik. The last completed issue of the Jahrbuch, that for 1889, extends to upwards of 1300 pages, while that for 1890, two parts of which have appeared, will evidently be as large.