BY FRANCIS CUTHBERTSON, M.A. KIANT 30 London : 1874. PREFACE. BEFORE entering into a detailed account of the plan of the present work one characteristic may perhaps deserve especial prominence, namely that, while all those parts of the Elements of Euclid which are required by the Universities have been established and at the same time Problems separated from Theorems, both classified according to the nature of the subject, and demonstrations replaced by others less cumbrous, I have been careful in all cases to retain Euclid's order as far as is necessary to allow of the proofs given being substituted for those of Euclid in Examinations. How important this is must be at once apparent to all who are conversant with Examinations either at Schools or Universities. One of the most generally acknowledged defects in the Elements of Euclid is the treatment of parallels; for a theorem is assumed as self-evident which certainly requires proof as much as those which are made to depend upon it. To avoid this difficulty a Lemma has been introduced which, while it rests on an axiomatic basis, brings out the property of the parallelism (or in common language equidistance) of straight lines in the same plane which never meet. This is of great consequence, as the leading idea in connection with straight lines called parallel is that suggested by the literal meaning of the word: thus the conception of a parallelogram is not that of a figure whose opposite sides will never meet however far they may be produced in either direction, but rather that of a figure whose opposite sides are equidistant Again, Euclid's treatment of ratio and proportion is objected to as being only adapted to the grasp of the advanced geometrician; one of the great difficulties to be encountered by a student attempting to master Euclid's Fifth Book being that magnitudes in general are dealt with instead of certain specified magnitudes. Accordingly to supply the want due to the fact of this Book being invariably omitted, the necessary propositions contained in it have been established for those particular species of magnitude to which they are afterwards applied. The method given possesses the combined advantages of being rigorous, geometrical, simple and short. The terms duplicate ratio and compound ratio have been relegated to the Supplement, there placed for the benefit of those likely to be examined thereon, but ejected from the text, as not being generally appreciated by students, few of whom would estimate the relative magnitudes of plane similar rectilineal figures by contemplating their homologous sides and then calculating their duplicate ratio. In order to make the Second Book, which treats of the relations existing between the rectangles under the segments |