From the Preface to the First Edition (1906): ``There are no definitely accepted landmarks in the didactic treatment of Georg Cantor'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 ... But we may fairly claim that the present work is the first attempt at a systematic exposition of the subject as a whole.'In this second edition, notes have been added by I. Grattan-Guinness drawn from extensive annotations in the author's own copy. A further appendix has been added.