The present English edition is not a mere translation of the German
original. Many new problems have been added and there are also other
changes, mostly minor. Yet all the alterations amount to less than ten
percent of the text. We intended to keep intact the general plan and the
original flavor of the work. Thus we have not introduced any essentially
new subject matter, although the mathematical fashion has greatly
changed since 1924. We have restricted ourselves to supplementing the
topics originally chosen. Some of our problems first published in this
work have given rise to extensive research. To include all such
developments would have changed the character of the work, and even an
incomplete account, which would be unsatisfactory in itself, would have
cost too much labor and taken up too much space. We have to thank many
readers who, since the publication of this work almost fifty years ago,
communicated to us various remarks on it, some of which have been
incorporated into this edition. We have not listed their names; we have
forgotten the origin of some contributions, and an incomplete list would
have been even less desirable than no list. The first volume has been
translated by Mrs. Dorothee Aeppli, the second volume by Professor
Claude Billigheimer. We wish to express our warmest thanks to both for
the unselfish devotion and scrupulous conscientiousness with which they
attacked their far from easy task.