ALGEBRA is the logic of Arithmetic. It is an abstraction of those operations, of which Arithmetic is an exemplification of particulars. He who understands Algebra must understand Arithmetic, of which science it illustrates the reason of every operation. Hence, Algebra is one of the most important studies, at once useful and elegant, which can be pursued during the period of education.' Of this truth, every one acquainted with Algebra must be convinced. It is the most refined species of logic or universal reasoning; an expert Algebraist cannot fail to be an acute reasoner, not only in the affairs of common life, but in all pursuits of Literature and Philosophy. Style itself is characterized by an acquaintance with this science. The language of the Algebraist is more terse, his arguments more relevant, and his conclusions powerful and convincing; on the other hand, the style of one who is ignorant of the art, is loose or turgid, and devoid of that influence on the mind, which the Algebraist always produces with dexterity and certainty. If so important an art has been less taught than it deserves to be in the ordinary Education of British youth, the cause appears to be the want of a neat and compact Elementary Treatise, adapted to the practical use of the schoolmaster, and also, to that economy of price which schoolmasters are obliged to respect, for the satisfaction of parents. A 1 Both these purposes will, it is presumed, be found to be effected by the present work; the Author is himself a schoolmaster, and a teacher of the mathematics of nearly twenty years experience; and having in his profession had the good fortune to acquire a fair reputation as a teacher, in one of the most populous towns in England, he may be presumed to be qualified for the task which he has undertaken. None but schoolmasters can estimate a schoolmaster's difficulties, and few can have a chance of removing them but those who have felt them. It is this circumstance which emboldens the Author of this treatise to calculate on its success. He affects to have made no discoveries, but cherishes the useful ambition of rendering easy and practicable, the acquirement of a useful and necessary branch of knowledge. It need scarcely be remarked to intelligent teachers, that before a youth commences this study, he ought to be moderately acquainted with the Elements of Arithmetic, and especially versed in the working of Fractions. These facilities may be acquired by any system of Arithmetic which may be agreeable to the master, for Algebra is a science which grows out of the general principles of Arithmetic, and its study may follow any method by which those principles are taught. Classical, Commercial, and Mathematical Academy, Trafalgar-street, Leeds, January 21st, 1829.) · PREFACE. ALGEBRA is the logic of Arithmetic. It is an abstraction of those operations, of which Arithmetic is an exemplification of particulars. He who understands Algebra must understand Arithmetic, of which science it illustrates the reason of every operation. Hence, Algebra is one of the most important studies, at once useful and elegant, which can be pursued during the period of education." Of this truth, every one acquainted with Algebra must be convinced. It is the most refined species of logic or universal reasoning; an expert Algebraist cannot fail to be an acute reasoner, not only in the affairs of common life, but in all pursuits of Literature and Philosophy. Style itself is characterized by an acquaintance with this science. The language of the Algebraist is more terse, his arguments more relevant, and his conclusions powerful and convincing; on the other hand, the style of one who is ignorant of the art, is loose or turgid, and devoid of that influence on the mind, which the Algebraist always produces with dexterity and certainty. If so important an art has been less taught than it deserves to be in the ordinary Education of British youth, the cause appears to be the want of a neat and compact Elementary Treatise, adapted to the practical use of the schoolmaster, and also, to that economy of price which schoolmasters are obliged to respect, for the satisfaction of parents. A Both these purposes will, it is presumed, be found to be effected by the present work; the Author is himself a schoolmaster, and a teacher of the mathematics of nearly twenty years experience; and having in his profession had the good fortune to acquire a fair reputation as a teacher, in one of the most populous towns in England, he may be presumed to be qualified for the task which he has undertaken.— None but schoolmasters can estimate a schoolmaster's difficulties, and few can have a chance of removing them but those who have felt them. It is this circumstance which emboldens the Author of this treatise to calculate on its success. He affects to have made no discoveries, but cherishes the useful ambition of rendering easy and practicable, the acquirement of a useful and necessary branch of knowledge. It need scarcely be remarked to intelligent teachers, that before a youth commences this study, he ought to be moderately acquainted with the Elements of Arithmetic, and especially versed in the working of Fractions. These facilities may be acquired by any system of Arithmetic which may be agreeable to the master, for Algebra is a science which grows out of the general principles of Arithmetic, and its study may follow any method by which those principles are taught. Classical, Commercial, and Mathematical Academy, Trafalgar-street, Leeds, January 21st, 1829., DEFINITIONS. 1. The Sign (plus) signifies that the quantity to which it is prefixed, is to be added. Thus a + x, (pronounced a plus x) denotes that x, or the number or quantity which it represents, is to be added to a. 2. The Sign (minus) signifies that the quantity to which it is prefixed, is to be subtracted. Thus a-x (pronounced a minus x) denotes that x, or the number which it represents, is to be subtracted from a. N. B. The quantities to which the Sign + is prefixed are called positive, and those to which the Sign is prefixed are called negative quantities. But a quantity without a Sign is always positive. 3. The Sign x (multiplication) signifies that the quantities between which it stands, are to be multiplied together. Thus a xx, (pronounced a multiplied by x, or a into x) makes ax. The product of single letters is expressed by writing them down one after the other. Thus, the product of a, b, and c, is abc, or it may be, bca, or cba, or acb, or bac, or cab, the product not being affected by taking any of the letters first-see the following example in figures: Let it be required to find the product of 4 and 6? Here the product is the same when 4 is made the multiplicand and 6 the multiplier, as when 6 is made the multiplicand and 4 the multiplier. |