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 volume contains a translation of the two very important memoirs of Georg Cantor on transfinite numbers which appeared in the Mathematische Annalen for 1895 and 1897 under the title: "Beiträge zur Begründung der transfiniten Mengenlehre." It seems to me that, since these memoirs are chiefly occupied with the investigation of the various transfinite cardinal and ordinal numbers and not with investigations belonging to what is usually described as "the theory of aggregates" or "the theory of sets" (Mengenlehre, théorie des ensembles)—the elements of the sets being real or complex numbers which are imaged as geometrical "points" in space of one or more dimensions,—the title given to them in this translation is more suitable.

These memoirs are the final and logically purified statement of many of the most important results of the long series of memoirs begun by Cantor in 1870. It is, I think, necessary, if we are to appreciate the full import of Cantor's work on transfinite numbers, to have thought through and to bear in mind Cantor's earlier researches on the theory of point-aggregates. It was in these researches that the need for the