' IлНК essays which comprise this volume appeared first in The •*■ Monist at different times during the years 1905 to 1916, and under different circumstances. Some of the diagrams were photographed from the authors' drawings, others were set in type, and different authors have presented the results of their labors in different styles. In compiling all these in book form the original presentation has been largely preserved, and in this way uniformity has been sacrificed to some extent. Clarity of presentation was deemed the main thing, and so it happens that elegance of typographical appearance has been considered of secondary importance. Since mathematical readers will care mainly for the thoughts presented, we hope they will overlook the typographical shortcomings. The first edition contained only the first eight chapters, and these have now been carefully revised. The book has been doubled in volume through the interest aroused by the first edition in mathematical minds who have contributed their labors to the solution of problems along the same line.