THE NEW YORK 686970 1915 Entered, according to Act of Congress, in the year 1851 by HARPER & BROTHERS, In the Clerk's Office of the Southern District of New York. PREFACE. Having understood that my “ Treatise on Algebra," which was designed primarily for the use of colleges, has been introduced into many academies and high schools, and employed in the instruction of classes younger than those for whom it was originally prepared, I have thought that a more elementary work, expressly designed for beginners, might be favorably received. The present volume was intended for the use of students who have just completed the study of Arithmetic; and it is believed that any person, how. ever young, who has acquired a tolerably thorough knowledge of the principles of that science, may proceed at once to this volume with pleasure and profit. I have endeavored to render the transition from Arith. metic to Algebra both easy and natural. This I have done by applying the algebraic symbols to problems 80 simple that they might be readily solved by the principles of Arithmetic alone. Having conducted the student through a considerable series of simple problems, I proceed, by easy steps, to develop some of them in a more general form. The student is thus Elod to represent known as well as unknown quantities TRANSFER FROM C. D. JUN by letters of the alphabet, and perceives the necessity of establishing rules for performing the various operations of addition, subtraction, multiplication, and division upon quantities thus represented. It is believed that the beginner will study these abstract principles with more satisfaction, than if he had been allowed no previous exercise on problems which indicate their importance. I have omitted from this volume all such topics as it was supposed would occasion any serious embarrassment to the young learner, and which were not essential to the clear comprehension of the topics actually introduced. It is hoped that the book will be found sufficiently clear and simple to be adapted to the wants of a large class of students in our common schools. The study of Algebra may properly be commenced at an early stage of education. As soon as the mind has acquired some degree of maturity, and has become familiar with the principles of common Arithmetic, it is prepared to understand the elementary principles of Algebra. This study is admirably adapted to strengthen the reasoning faculties, to lead the mind to rely upon its own resources, and to cultivate those habits of independent thinking which are alike important to the scholar and to the man of business CONTENTS. PRELIMINARY DEFINITIONS AND FIRST PRINCIPLES. Signs of Addition, Subtraction, etc.. Exercises upon the Symbols............ Definitions of Equation, Terms, etc. .... Mode of solving a Problem ......... Definitions of Positive and Negative Quantities Radical Sign-Index of a Root... Three Cases distinguished....... Arithmetical and Algebraic Addition..... Arithmetical and Algebraic Subtraction ... Rule for the Exponents ......... Rule for the Signs............ Quantities resolved into Factors...... General Rule .......... To reduce a Fraction to Lower Terms ... To reduce a Fraction to an Entire Quantity .. To reduce a Mixed Quantity to the Form of a Fraction.... To reduce Fractions to a Common Denominator .. To add Fractional Quantities together ....... To subtract one Fraction from another........ EQUATIONS OF THE FIRST DEGREE. Equations solved by Addition and Subtraction ....... Equations solved by Multiplication and Division . Equation cleared of Fractions.... .... |