NEW UNIVERSITY ARITHMETIC, EMBRACING THE SCIENCE OF NUMBERS, AND THEIR APPLICATIONS ACCORDING TO THE MOST IMPROVED METHODS OF ANALYSIS AND CANCELLATION. BY CHARLES DAVIES, LL. D., AUTHOR OF PRIMARY, INTELLECTUAL, AND SCHOOL ARITHMETICS; ELEMENTARY NEW YORK: PUBLISHED BY A. S. BARNES & CO., 1857. www Entered according to act of Congress, in the year eighteen hundred and fifty-six, BY CHARLES DAVIES, In the Clerk's Office of the District Court of the United States, for the Southern District of New York. JONES & DENYSE, STEREOTYPERS AND ELECTROTYPERS, 183 William-Street. PREFACE. SCIENCE, in its popular signification, means knowledge reduced to order; that is, knowledge so classified and arranged, as to be easily remembered, readily referred to, and advantageously applied. ARITHMETIC is the science of numbers. It is the foundation of the exact and mixed sciences and a knowledge of it is an important element either of a liberal or practical education. While Arithmetic is a science in all that concerns the properties of numbers, it is an art in all that relates to their practical application. It is the first subject in a well-arranged course of instruction to which the reasoning powers of the mind are applied, and is the guide-book of the mechanic and man of business. It is the first fountain at which the young votary of knowledge drinks the pure waters of intellectual truth. It has seemed, to the author, of the first importance that this subject should be well treated in our Elementary Text Books. In the hope of contributing something to so desirable an end, he has prepared a series of arithmetical works, embracing four books, entitled, Primary Arithmetic; Intellectual Arithmetic; School Arithmetic; and University Arithmetic-the latter of which is the present volume. PRIMARY ARITHMETIC. This first book is adapted to the capacities and wants of young children. Sensible objects are employed to illustrate and make familiar the simple combinations and relations of numbers. Each lesson embraces one combination of numbers, or one set of combinations. INTELLECTUAL ARITHMETIC. This work is designed to present a thorough analysis of the science of numbers, and to form a complete course of mental arithmetic. It is thought to be accessible to young pupils by the simplicity and gradation of its methods, and to be particularly adapted to the wants of advanced students, as the attempt has been faithfully made to give the subjects of which it treats a scientific arrangement and logical connection in all the higher methods of arithmetical analysis. SCHOOL ARITHMETIC. Great pains has been taken in the preparation of this book to combine theory and practice; to explain and illustrate principles, and to apply them to the common business transactions of life-to make it emphatically a practical work. The student is required to demonstrate every principle laid down, by a course of mental reasoning, before deducing a proposition or making a practical application of a rule to examples. He is required to fix upon the unit or unity as the base of all numbers, whether integral or fractional to reason with constant reference to this base, and thus make it the key to the solution of all arithmetical questions. It is thought, that the language used in the statement of principles, in the definitions of terms, and in the explanation of methods, will be found to be clear, exact, brief and comprehensive. UNIVERSITY ARITHMETIC. This work is designed to answer another object. Here, the entire subject is treated as a science. The scholar is supposed to be familiar with the simple operations in the four ground rules, and with the first principles of fractions, these being now taught to small children either orally or from elementary treatises. This being premised, the language of figures, which are the representatives of numbers, is carefully taught, and the different significations of which the figures are susceptible, depending on the manner in which they are written, are fully explained. It is shown, for example, that the simple numbers in which the value of the unit |