PREFATORY NOTE TO THE NEW EDITION. In the present edition the text has received further revision, and the notes have been for the most part re-written, with a view to greater clearness and simplicity. References to the Definitions being frequent in the text of Book I., the convenience of a standard order has been pointed out to us by many elementary teachers. We have therefore thought it advisable to re-number the Definitions in accordance with Simson's edition. This has involved the insertion of certain definitions hitherto omitted as of slight importance : such insertions have now been printed in subordinate type. A few typographical improvements have been introduced : notably the italicising of Particular Enunciations. Some changes in pagination have also been effected for the purpose of presenting the whole of a proposition at one view, or of bringing notes and exercises into closer connection with the text to which they refer. Further, the symbols “: " for therefore, ” for is equal to are now introduced from the 35th Proposition of Book I. Groups of Test Questions for Revision have been inserted at various stages. These may be useful to beginners, and suggestive to teachers in framing examination papers, which so often consist of mere monotonous lists of propositions and examples. One important change has been made. The algebraical treatment of the subject-matter of Book V. has been entirely separated from the stricter general treatment, so as to present in the simplest form such Definitions and Theorems of Proportion as are necessary before entering upon Book VI. This Introduction will be found immediately preceding Book VI. in a chapter called The Elementary Principles of Proportion. H. S. H. and " February, 1900. CONTENTS DEFINITIONS, POSTULATES, AXIOMS SECTION I. PROPOSITIONS 1-26 . SECTION II. PARALLELS AND PARALLELOGRAMS. Theorems and Examples on Book III. I. ON THE CENTRE AND CHORDS OF A CIRCLE II. ON THE TANGENT AND THE CONTACT OF CIRCLES. The Common Tangent to Two Circles, Problems on III. ON ANGLES IN SEGMENTS, AND ANGLES AT THE CEN- TRES AND CIRCUMFERENCES OF CIRCLES. The Orthocentre of a Triangle, and Properties of the Pedal Triangle, Loci, Simson's Line The Nine-Points Circle II. MISCELLANEOUS EXAMPLES 297 303 Theorems and Examples on Book VI. II. ON CENTRES OF SIMILARITY AND SIMILITUDE The First Proposition of Book XII. will be found at p. 364, worked out as an Example on Proposition 20 of Book VI., |