THE ELEMENTS OF ALGEBRA. BY ARCH MONTGOMERIE, TEACHER OF MATHEMATICS, ETC. IN AYR ACADEMY. BODL OTHE EDINBURGH: PUBLISHED FOR THE AUTHOR, BY OLIVER & BOYD, TWEEDDALE COURT : SIMPKIN, MARSHALL, & CO., LONDON. 1857. 181. c. 122. PREFACE. THE following pages have been compiled chiefly for the use of the Author's own Classes, and it has been his object to supply a useful and sufficiently comprehensive elementary work at as moderate a price as possible. With this view, the long and somewhat abstract demonstrations of the Rules and Principles employed in the Fundamental operations have been omitted, partly because they may be much better given orally, as circumstances require, and partly because they are totally unintelligible to a pupil of ordinary capacity until he has acquired some practical knowledge of the operations themselves, and a certain degree of facility in performing them. Most of these demonstrations, however, the learner is afterwards required to supply himself, in the form of Exercises, so as to cultivate his powers of analytical investigation, and train his mind to habits of correct thought. As the work progresses, and when the pupil may be supposed to be prepared for it, the Theoretic element is gradually introduced, and occupies the prominent place which it ought to do in every class-book. The Exercises have been carefully prepared and arranged; some of them are selected from the best London and Cambridge works, and a considerable number from the Exami nation Papers for appointments to the Royal Artillery, and for admission to the Royal Military Academy at Woolwich. The greater part of these last have been placed at the end of Chapter XIII., and will be found to be by no means so difficult as many of the others in that chapter. A thorough acquaintance with what is here given will therefore form, so far as Algebra is concerned, an ample preparation for these important Examinations, in which, as is well known, Mathematical science occupies the first place. H ALGEBRA.. CHAPTER I. DEFINITIONS. 1. IN Algebra, the quantities under consideration are expressed by means of the letters of the alphabet. Known quantities are usually expressed by the first letters, as a, b, c, &c., and unknown quantities by the last, as x, y, z. Sometimes the letters of the Greek alphabet are employed, and sometimes also the capitals A, B, C, &c., according to circumstances. 2. The sign + (plus) is the sign of Addition, and denotes that the quantity to which it is prefixed must be added. Such quantities are called positive. 3. The sign (minus) is the sign of Subtraction, and denotes that the quantity to which it is prefixed must be subtracted. Such quantities are called negative. Thus, if a represent 7, and b, 3; then a + b represents 10, and a-b, 4. NOTE. When no sign is prefixed, + is always understood. 4. To denote the multiplication of quantities, the letters representing them are usually written in succession, as in a word. Thus ab denotes that the quantity represented by a is to be multiplied by the quantity represented by b. In some cases a point, or the sign x (into) is used to denote Multiplication. Thus 2.3.4, or 2 x 3 x 4, denotes the continued product of 2, 3, and 4. 5. If the same quantity be repeated as a factor any number of times, the product is called a power of that quantity. A |