The following lectures contain an elementary account of the logical foundations of algebra and geometry, -- elementary, in the sense that the technical mathematical equipment presupposed on the part of the reader has been reduced to a minimum. Except in a very few instances, no knowledge of mathematics beyond the most elementary portions of algebra and geometry has been assumed. It has been my purpose to give a general exposition of the abstract, formal point of view developed during the last few decades, rather than an exhaustive treatment of the details of the investigations.

The results of recent work on the logical foundations are of vital interest alike to the teachers of mathematics in our secondary schools and colleges and to philosophers and logicians. I hope that both these classes will welcome a concise statement of some of the more fundamental of these results and an elementary exposition, omitting all involved details, of the point of view which governs all present work on the foundations. The book should be available also as a text in connection with so-called Teachers' Courses in colleges and universities.