PREFACE. IT IS HOPED that the title which has been assumed for this publication will be justified by the success of the work in meeting the requirements of the present day. An endeavour has been made to include in one book all modern improvements in their most matured and useful forms, and so to teach and exemplify a general conciseness of method in the solution of problems, as to promote greater facility and power of comprehension in the pupil's mind than the tedious methods that are still so prevalent. The problems themselves, which are very numerous, and nearly all original, are carefully graduated; even those under the head of Miscellaneous Exercises are divided into brief sets, each of which forms a progressively arranged Examination Paper; while several of a kind which has hitherto been found only among miscellaneous questions are, by very simple demonstration, reduced to one Rule. With regard to the order of the Rules in this Treatise, the author has preferred expedient arrangement to that which according to abstract system may be regarded as more scientific; but as the principal diffi culties in Arithmetic arise from the peculiarities of particular examples, an abundant variety of examples are here worked out, with an amount of explanation sufficient to promote in the pupil a proper aptitude for the solution of similar problems. Those arithmetical students who take pleasure in trying to solve difficult questions will find here, as well as in the author's Appendix to Colenso's Arithmetic, ample scope for the exercise of their 11 while those who |