The present volume is an attempt at a simple presentation of one of the most recent branches of mathematical science. It has involved an amount of labour which would seem to the average reader quite out of proportion to the size of the book; yet I can scarcely hope that the mode of presentation will appeal equally to all mathematicians. There are no definitely accepted landmarks in the didactic treatment of Georg Cantorg's magnificent theory, which is the subject of the present volume. A few of the most modern books on the Theory of Functions devote some pages to the establishment of certain results belonging to our subject, and required for the special purposes in hand. There is moreover in existence the first half of Schoenflies's useful Bericht über die Mengenlehre. The philosophical point of view is discussed to some extent in Russell's Principles of Mathematics. But we may fairly claim that the present work is the first attempt at a systematic exposition of the subject as a whole.

The difficulties in arrangement which this fact suggests have been enhanced by the nature of the subject itself and by the tentative character of some of its results. The writing of the book has necessarily involved attempts to extend the frontier of existing knowledge, and to fill in gaps which broke the connexion between isolated parts of the subject.