PREFACE. THE PURPOSE of this book is to explain the rudiments of Algebra and Trigonometry to artisans and others, who may wish to be acquainted with them so far as to make the computations which arise in practice, and to read books in which science is treated mathematically. It is my hope that a student who masters this book and works its examples will find himself able to solve a large number of the questions which applied science raises, and to perform all the ordinary calculations which logarithms assist, and that he will find himself in possession of a trustworthy and available power, although, so far as I conduct him, he will have entered but a small portion of the wide field of modern analysis. The Examples in this book are taken, with a few exceptions, from examination papers publicly accessible. The reader who has no previous knowledge of the subjects of this work, is recommended to omit for a time the following articles : Algebra.-123-125. 255-260. 262-287. Trigonometry.-Chap. IV. 117-119. 123-127. 130-132. 134 to the end of the chapter. 156 to the end. OSPRINGE VICARAGE: January 4, 1871. W. N. GRIFFIN. ALGEBRA. CHAPTER I. THE MEANING OF ALGEBRAICAL SYMBOLS. 1. The method taken in this book is to conduct the reader into Algebra by regarding it as an extension of Arithmetic. The principles and processes of Arithmetic are therefore supposed to be already well known. ALGEBRAIC SYMBOLS AND THEIR GENERALITY. 2. In Arithmetic, it will be remembered that symbols are used, digits as they are called, which have a certain generality, and mean different things as the unit is varied. The symbol 8, for instance, means eight units, but it may be eight shillings, eight pounds, eight yards, as may be intended. Operations are performed with these symbols, and true results are obtained, whatever be the unit in view. Eight and five added together make thirteen, whether they be pounds, or gallons, or inches which are thus being added together. Now in Algebra representations of quantity are taken which have a further generality. Quantity is designated by a symbol, most commonly a letter of the alphabet, a, b, c, m, x, for instance, and now not only the unit, but the number of units signified, waits to be assigned. The lettera, for instance, may represent any number referred to any unit. Operations will be performed with these letter representatives of quantity, and thereby problems will be solved for which the processes of arithmetic are insufficient. 3. Obs. The word number in this book is to be taken to mean either an integer, or a quantity wholly or partly decimal. 4. Out of this statement of the purposes and method of algebra, many questions may at once arise in the reader's mind, how things so arbitrary as letters, so unconnected in themselves with magnitude, can have any definiteness, and can lead to any certain numerical results, how such symbols can be kept from confusion, how we are to know what each means. Such questions as these are noticed to assure the reader that his probable difficulties are foreseen and acknowledged, but they are questions which cannot be answered as yet. They will cease to give any difficulty as soon as algebra is seen in its application to practical questions. In the earlier pages of any treatise on algebra, the reader commencing the subject ought not to be surprised or discouraged if he sees but indistinctly the use of the processes in which he is being instructed. He will not be required, it is hoped, to accept any result without sufficient proof, but he may not see for a while the objects and purposes of the statements demonstrated. 5. Other symbols besides the italic letters of the alphabet are brought into use in algebra, to designate quantity. Affixes are sometimes attached to a letter, as a1, a, a,...making thereby so many different symbols. The Greek alphabet, a, ß, y,...is also brought into service. A symbol of quantity in algebra is any mark which can be recognised and reproduced. 6. Def.-gebraical symbols written down as a representation of magnitude, form what is termed an expression.' |