AN ELEMENTARY TREATISE ΟΝ 17-13.9 ALGEBRA. DESIGNED AS FIRST LESSONS IN THAT SCIENCE, BY H. N. ROBINSON, A. M., AUTHOR OF AN UNIVERSITY EDITION OF ALGEBRAAN ELEMENTARY TREATISE ON ASTRONOMY, AND SEVERAL OTHER MATHEMATICAL WORKS. 1 SECOND EDITION. CINCINNATI: JACOB ERNST NO. 183, MAIN STREET ALBANY : ERASTUS H. PEASE & CO. 1850. Entered according to Act of Congress in the year 1850, by. H. N. ROBINSON, In the Clerk's Office of the District Court of the United States, for the District of Ohio. STEREOTYPED BY JAMES & CO., CINCINNATI. PREFACE. EVERY teacher is desirous of having as few textbooks in his school as is consistent with efficient and sound instruction, and in accordance with this object, great efforts have been made by several authors to produce a work on Algebra that would be a proper textbook for all grades of pupils. But in this all have failed, and given up the point in despair. The student of adult age, and possessing a passably disciplined mind, requires a different book from the mere lad, who is just commencing the science. If we put a child's book into the hands of a young man, he will, probably, become displeased with the book, and possibly imbibe prejudice and distaste for the science itself; and if we put a logical and philosophical work into the nands of a child, he is sure not to comprehend it, however well and fluently he may be made to repeat the contents of its pages. But, nevertheless, as Algebra is the groundwork of all the mathematical sciences, and is of itself a system of pure logic, it is important that it should be commenced at an early age-eleven or twelve, or if otherwise well employed, thirteen or fourteen is a more suitable age, It is a prevalent impression that Algebra should not be commenced until the pupil has acquired a gocd knowledge of Arithmetic, but this is a great error. The impression would be well founded, provided Arithmetic was the most elementary science, and Algebra was founded on Arithmetic; but the reverse is the fact-Algebra is elementary Arithmetic, and no one can acquirc & knowledge of Arithmetic in an enlarged and scientific sense, without a previous knowledge of Algebra. Beyond notation, numeration, and the four simple rules, Arithmetic is not a science, but a sequel to all sciences, it is numerical computation applied to anything and to everything. Proportion, as a science, is the comparison of magnitudes, and belongs, properly speaking, to Algebra and Geometry; and the rule of three, in Arithmetic, is but little more than some of its forms of application. Problems in mensuration are very properly to be found in books called Arithmetics, but mensuration is no part of the science of Arithmetic, it is a part of Geometry, and for a good understanding of it, geometrical science must be directly consulted. So it is with many other parts of Arithmetic, the science is elsewhere; and to have a scientific comprehension of many parts of common Arithmetic, we must go to general Arithmetic, which is emphatically Algebra; and in preparing this work, we have given constant attention to this branch of the subject, as may be seen in our treatment of fractions, proportion, progression, the roots, fellowship, and interest. All these subjects can be better illustrated by symbols than by numbers; for numbers apply to everything, and, of course, can be made to show no particular thing; but not so with symbols, at every step the particular elements are all visible, and the logic and the reason is as distinct in every part of an operation as is the result. For these reasons, Arithmetic should be studied by symbols, as it is in many parts of Europe; many of their books, entitled Arithmetics, are as full of signs and symbols as any Algebra that ever appeared. The prominent design of the author has been to adapt this treatise to the wants of young beginners in Algebra, and at the same time not to produce a mere childish book, but one more dignified and permanent, and to secure this end, he has kept up the same tone and spirit as though he were addressing mature and disciplined minds. Great care has been taken in the selection of problems, and all very severe ones have been excluded, and all such as might be difficult when detached and alone, are rendered simple and easy by their connection with other leading problems of kindred character. To bring out the original thoughts of the pupil has been another object which he designed to accomplish, and the illustrations are given in such a way as to command the constant attention of the learner, and if he learns at all, it will be naturally and easily, and what he learns will, become a part of himself. In this work, great importance is attached to equations, not merely ⚫ in solving problems, but they are used as an instrument of illustrating principles, and their application is carried further in this book than in .any other known to the author. For instance, we have illustrated the nature of an equation by the aid of simple problems in subtraction and division; and conversely, the simple principle of equality is used to deduce rules for subtraction, division, the reduction of fractions to a common denominator, the multiplication of quantities affected by different fractional exponents, &c. Notwithstanding that this book is designed to be practical, it contains more illustrations, and is more theoretical and scientific as far as it goes, than any other book designed for the same class of pupils. |