JOHN CASEY, ESQ., LL.D., F. R. S., FELLOW OF THE ROYAL UNIVERSITY OF IRELAND. Fourth Edition, Revised and Enlarged, Price 38. A TREATISE ON ELEMENTARY TRIGONOMETRY, With numerous Examples AND Questions for Examination. Second Edition, Price 38. KEY TO THE EXERCISES IN THE TREATISE ON ELEMENTARY TRIGONOMETRY. Seventh Edition, Revised and Enlarged, Price 38. 6d., Cloth. Containing an Easy Introduction to Modern Geometry, Fourteenth Edition, Price 4s. 6d. THE ELEMENTS OF EUCLID, BOOKS I.-VI., AND PROPOSITIONS I.-XXI. OF BOOK XI.; Together with an Appendix on the Cylinder, Sphere, Cone, &c., Copious Annotations & numerous Exercises. Third Edition, Price 6s. A KEY TO THE EXERCISES IN THE FIRST SIX BOOKS OF CASEY'S "ELEMENTS OF EUCLID." Second Edition, Price 12s. A TREATISE ON THE ANALYTICAL GEOMETRY OF THE POINT, LINE, CIRCLE, & CONIC SECTIONS, Containing an Account of its most recent Extensions, With numerous Examples. Second Edition (in preparation), Price 7s. 6d. A TREATISE ON PLANE TRIGONOMETRY, including THE THEORY OF HYPERBOLIC FUNCTIONS, With numerous Examples. Second Edition (in preparation), Price 58. A TREATISE ON SPHERICAL TRIGONOMETRY, and its APPLICATIONS to GEODESY and ASTRONOMY, With numerous Examples. LONDON: LONGMANS & CO. DUBLIN: HODGES, FIGGIS, & CO. (LTD.). ABGould DUBLIN UNIVERSITY PRESS SERIES. A SEQUEL TO THE FIRST SIX BOOKS OF THE ELEMENTS OF EUCLID, CONTAINING AN EASY INTRODUCTION TO MODERN GEOMETRY, BY JOHN CASEY, LL.D., F. R.S., Fellow of the Royal University of Ireland; Member of the Council of the Royal Irish Academy; DUBLIN: HODGES, FIGGIS, & CO. (Lîn.), GRAFTON-ST. PREFACE. I HAVE endeavoured in this Manual to collect and arrange all those Elementary Geometrical Propositions not given in Euclid which a Student will require in his Mathematical Course. The necessity for such a Work will be obvious to every person engaged in Mathematical Tuition. I have been frequently obliged, when teaching the Higher Mathematics, to interrupt my demonstrations, in order to prove some elementary Propositions on which they depended, but which were not given in any book to which I could refer. The object of the present little Treatise is to supply that want. The following is the plan of the Work. It is divided into five Chapters, corresponding to Books I., II., III., IV., VI. of Euclid. The Supplements to Books I.-IV. consist of two Sections each, namely, Section I., Additional Propositions; Section II., Exercises. This part will be found to contain original proofs of some of the |