PREFACE. THE present work has been undertaken at the request of many teachers, in order to be placed in the hands of beginners, and to serve as an introduction to the larger treatise published by the author; it is accordingly based on the earlier chapters of that treatise, but is of a more elementary character. Great pains have been taken to render the work intelligible to young students, by the use of simple language and by copious explanations. In determining the subjects to be included and the space to be assigned to each, the author has been guided by the papers given at the various examinations in elementary Algebra which are now carried on in this country. The book may be said to consist of three parts. The first part contains the elementary operations in integral and fractional expressions; it occupies eighteen chapters. The second part contains the solution of equations and problems; it occupies twelve chapters. The subjects contained. in these two parts constitute nearly the whole of every examination paper which was consulted, and accordingly they are treated with ample detail of illustration and exercise. The third part forms the remainder of the book; it consists of various subjects which are introduced but rarely into the examination papers, and which are therefore more briefly discussed. The subjects are arranged in what appears to be the most natural order. But many teachers find it advan tageous to introduce easy equations and problems at a very early stage, and accordingly provision has been made for such a course. It will be found that Chapters XIX. and XXI. may be taken as soon as a student has proceeded as far as algebraical multiplication. In accordance with the recommendation of teachers, the examples for exercise are very numerous. Some of these have been selected from the College and University examination papers, and some from the works of Saunderson and Simpson; many however are original, and are constructed with reference to points which have been shewn to be important by the author's experience as a teacher and an examiner. The author has to acknowledge the kindness of many distinguished teachers who have examined the sheets of his work and have given him valuable suggestions. Any remarks on the work, and especially the indication of difficulties either in the text or the examples, will be most thankfully received. I. TODHUNTER. ST JOHN'S COLLEGE, ERRATA. Page 60, Ex. 22, for +7 read – 7. Ex. 23, for x2+2x+3 read x2+2x-3. 65, CONTENTS. PAGE I. The Principal Signs II. Factor. Coefficient. Power. Terms..... IV. Change of the order of Terms. Like Terms V. Addition I 5 8 II 15 XXIII. Simultaneous Equations of the first degree with two unknown quantities XXIV. Simultaneous Equations of the first degree with more than two unknown quantities. XXV. Problems which lead to simultaneous equa- PAGE 134 143 147 XXVI. Quadratic Equations 158 XXVII. Equations which may be solved like Quad ratics 169 XXVIII. Problems which lead to Quadratic Equations 174 ratics XXX. Problems which lead to Quadratic Equations XXXI. Involution . 179 187 192 196 213 220 226 230 234 238 242 247 249 ALGEBRA FOR BEGINNERS. I. The Principal Signs. 1. ALGEBRA is the science in which we reason about numbers, with the aid of letters to denote the numbers, and of certain signs to denote the operations performed on the numbers, and the relations of the numbers to each other. 2. Numbers may be either known numbers, or numbers which have to be found, and which are therefore called unknown numbers. It is usual to represent known numbers by the first letters of the alphabet a, b, c, &c., and unknown numbers by the last letters x, y, z; this is however not a necessary rule, and so need not be strictly obeyed. Numbers may be either whole or fractional. The word quantity is often used with the same meaning as number. The word integer is often used instead of whole number. 3. The beginner has to accustom himself to the use of letters for representing numbers, and to learn the meaning of the signs; we shall begin by explaining the most important signs and illustrating their use. We shall assume that the student has a knowledge of the elements of Arithmetic, and that he admits the truth of the common notions required in all parts of mathematics, such as, if equals be added to equals the wholes are equal, and the like. T. A. 1 |