”In the text of the propositions, the author has adhered, in all but a few instances, to the substance of Euclid’s demonstrations, without, however, giving way to a slavish following of his occasional verbiage and redundance. The use of letters in brackets in the enunciations eludes the necessity of giving a second or particular enunciation, and can do no harm. Hints of other proofs are often given in small type at the end of a proposition, and, where necessary, short explanations. The definitions are also carefully annotated. The theory of proportion, Book V., is given in an algebraical form. This book has always appeared to us an exquisitely subtle example of Greek mathematical logic, but the subject can be made infinitely simpler and shorter by a little algebra, and naturally the more difficult method has yielded place to the less. It is not studied in schools, it is not asked for even in the Cambridge Tripos; a few years ago, it still survived in one of the College Examinations at St. John’s, but whether the reforming spirit which is dominant there has left it, we do not know. The book contains a very large body of riders and independent geometrical problems. The simpler of these are given in immediate connexion with the propositions to which they naturally attach; the more difficult are given in collections at the end of each book. Some of these are solved in the book, and these include many well-known theorems, properties of orthocentre, of nine-point circle, &c. In every way this edition of Euclid is deserving of commendation. We would also express a hope that everyone who uses this book will afterwards read the same author’s ‘Sequel to Euclid,’ where he will find an excellent account of more modern Geometry.” ________________ NOW READY, Price 6s., A KEY to the EXERCISES in the ELEMENTS of EUCLID. Typographical Errors corrected in Project Gutenberg edition p. ??. “Def. viii.—When a right line intersects …” in original, amended to “Def. vii” in sequence. p. ??. 12 “bisects the parallellogram” in original, amended to match every other occurrence as “parallelogram”. p. ??. “△ACH is half the rectangle AC.AH (I. Cor. 1)” in original. The reference is to Prop. I. of the current book and misnumbered, it should be (i. Cor. 2). p. ??. “The parallelogram CM is equal to DE [I. xliii., Cor. 3];” in original, amended to “Cor. 2” following MS. correction: there is no Cor. 3. p. ??. “On CB describe the square CBEF I. [xlvi.].” in original, clearly meant to read [I. xlvi.]. p. ??. “The remainiug parts of the line” in original, obvious error amended to “remaining”. p. ??. “that which is nearest to the line throuyh the centre” in original, obvious error amended to “through”. p. ??. “Then this line [I., Cor. 1]” in original. The reference is to Prop. I. of the current book, so it should be [i., Cor. 1]. p. ??. “OA is equal to OC [I., Def. xxii.]” in original. The reference should be [I., Def. xxxii.].