At the best, the work of the translator is a rather thankless task. It is a target for critics of style and for critics of matter. For the style of the German work the translators will hardly be held responsible. It is not a ?uent one, leaning too much to the scientific side to make it always easy reading. Were the work less scientific, it would lend itself more readily to a better English form, but the translators have preferred to err on the side of a rather strict adherence to the original. As to the matter, it has seemed unwise to make any consider able changes. The attempt has been made to correct a number of unquestionable errors, occasional references have been added, and the biographical notes have been rewritten. It has not seemed advisable, however, to insert a large number of bibliographical notes. Readers who are interested in the subject will naturally place upon their shelves the works of De Morgan, Allman, Gow, Ball, Heath, and other English writers, and, as far as may be, works in other languages. The leading German authorities are mentioned in the footnotes, and the French language offers little at present beyond the works of Chasles and Paul Tannery. The translators desire to express their obligations to Professor Markley for valuable assistance in the translation.