THE present work is an attempt to supply a want which we have long felt ourselves, and which we believe to be shared by many experienced teachers. In setting before a beginner the real and perplexing difficulties of elementary Algebra, there is some fear lest first lessons should degenerate into a mere mechanical manipulation of symbols, uninteresting and uninstructive, because little understood. This well known danger led us to devote special thought to the question of order; to consider, in short, what succession of the various parts of the subject would best illustrate its bearings at an early stage; and we have finally adopted an arrangement, which if it varies somewhat from the common use of elementary text-books-is at least based upon the experience of many years, and embodies the result of frequent consultation with our colleagues and other teachers. As regards the earlier chapters, our order has been determined mainly by two considerations: first, a desire to introduce as early as possible the practical side of the subject and some of its most interesting applications, such as easy equations and problems; and secondly, the strong opinion that all reference to compound expressions and their resolution into factors should be postponed until the usual operations of Algebra have been exemplified in the case of simple expressions. By this course the beginner soon becomes acquainted with the ordinary algebraical processes without encountering too many of their difficulties; and he is learning at the same time something of the more attractive parts of the subject. Again, by postponing Resolution into Factors until the student has acquired some freedom and readiness in the use of symbols, we are enabled to treat this important section, and all the processes to which it gives rise, more adequately than is possible where factors are introduced and disposed of in one short early chapter. We had originally intended to arrange the chapters in an order that might be followed without deviation from beginning to end; but in the course of the work this was found impossible without very extensive subdivision in some parts, and needless repetition in others. We have therefore marked with an asterisk all articles and examples which may conveniently be omitted by a student who is reading the subject for the first time, and we have occasionally added a note suggesting the most suitable place for a section which may have to be postponed. To one feature of the book we wish to draw special attention. After Quadratics we have introduced two recapitulatory chapters in which are discussed certain classes of questions requiring for their solution a very ready and skilful use of factors. Such questions are undoubtedly hard, and wholly out of place at an earlier stage; yet in existing text-books their importance is hardly recognised, and they are generally introduced at a point where the ordinary student cannot use or appreciate the best methods of solving them. To these harder applications of early rules in Chapters XXVIII and XXIX, we have added several useful miscellaneous theorems illustrated by carefully selected examples. These chapters are the outcome of our own experience in preparing boys for Army and University Examinations during the last twelve years; they will be found very valuable for revision, though they also contain some new and interesting matter. The examples are piled with great care. though some have been selected from recent Examination Papers. In the early chapters we have given an unusual number of very easy questions, many of which will be found well suited for viva voce work. In each chapter several examples are fully worked out to illustrate the most useful applications of important rules. Also, under the belief that prevalent mistakes are not sufficiently guarded against, we have given occasional notes to caution the reader against the blunders which experience shews to be almost universal among beginners. With the help of our colleagues we have endeavoured to give none but the best methods, so that students on passing from this to a more advanced treatise may have a good foundation, and have nothing to unlearn. very numerous, and have been comThey are for the most part original, We have done our best to secure accuracy in all parts of the book, but in a collection of 3500 examples it is probable that some mistakes still remain undetected: we shall be thankful to receive any corrections. We have to acknowledge our indebtedness to many kind friends for their valuable help and advice. In the arrangement of the early chapters we have been largely influenced by the opinion of the Rev. J. M. Wilson, Head Master of Clifton College; and practical suggestions received from him have been frequently worked into the text and examples throughout the book. We have also to express our thanks for the assistance and encouragement freely given by many of our Mathematical colleagues: to the Rev. H. C. Watson and Mr F. H. Stevens of Clifton College we are especially indebted for their practical hints and able criticism; while in revising the manuscript and correcting the proof sheets their aid has been most generous and untiring. Up to the end of Surds we hope the book will be found sufficiently complete; the remaining chapters have been introduced to meet the requirements of many Elementary Examinations, but they do not profess to give more than an easy first course of Ratio, Proportion, Variation, and the Progressions. These subjects will be treated more fully in the early chapters of our Higher Algebra, now in course of preparation, as a sequel to the present work. H. S. HALL, S. R. KNIGHT. June 1885. |