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The object of this treatise is to supply the want which is fell by Students of a suitable text-book on geometry. Hitherto the study of Pure Geometry has been neglected; chiefly, no doubt, because questions bearing on the subject have very rarely been set in examination papers. In the new regulations for the Cambridge Tripos, however, provision is made for the introduction of a paper on "Pure Geometry; - namely, Euclid; simple properties of lines and circles; inversion; the elementary proper of conic sections treated geometrically, not excluding the method of projections; reciprocation; harmonic properties, curvatu In the present treatise I have brought together all the important propositions - bearing on the simple properties of lines and circles - that might fairly be considered within the limits of the above regulation. At the same time I have endeavoured treat every branch of the subject as completely as possible in the hope that a larger number of students than at present may be induced to devote themselves to a science which deserves as much attention as any branch of Pure Mathematics.<br><br>Throughout the book a large number of interesting theorems and problems have been introduced as examples to illustrate the principles of the subject. The greater number have been taken from examination papers set at Cambridge and Dublin, or from the Educational Times. Some are original, while others are takei from Townsend's Modern Geometry, and Caseys Sequel to Euclid.