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vi transfinite numbers first showed itself, and it is only by the study of these researches that the majority of us can annihilate the feeling of arbitrariness and even insecurity about the introduction of these numbers. Furthermore, it is also necessary to trace backwards, especially through Weierstrass, the course of those researches which led to Cantor's work. I have, then, prefixed an Introduction tracing the growth of parts of the theory of functions during the nineteenth century, and dealing, in some detail, with the fundamental work of Weierstrass and others, and with the work of Cantor from 1870 to 1895. Some notes at the end contain a short account of the developments of the theory of transfinite numbers since 1897. In these notes and in the Introduction I have been greatly helped by the information that Professor Cantor gave me in the course of a long correspondence on the theory of aggregates which we carried on many years ago.

The philosophical revolution brought about by Cantor's work was even greater, perhaps, than the matheniatical one. With few exceptions, mathematicians joyfully accepted, built upon, scrutinized, and perfected the foundations of Cantor's undying theory; but very many philosophers combated it. This seems to have been because very few understood it. I hope that this book may help to make the subject better known to both philosophers and mathematicians.

The three men whose influence on modern pure mathematics—and indirectly modern logic and the