THE UNIVERSITY ARITHMETIC, EMBRACING THE SCIENCE OF NUMBERS, AND THEIR NUMEROUS APPLICATIONS RY CHARLES DAVIES, LL. D., AUTHOR OF FIRST LESSONS IN ARITHMETIC; ARITHMETIC; ELEMENTARY ALGEBRA : ELEMENTS OF SURVEYING; ELEMENTS OF ANALYTICAL GEOM- ETRY; DESCRIPTIVE GEOMETRY; SHADES, SHADOWS, AND PERSPECTIVE; AND DIFFERENTIAL AND INTEGRAL CALCULUS REVISED AND IMPROVED EDITION. NEW YORK: PUBLISHED BY A. S. BARNES & CO., No 51 JOHN STREET. 1850. Entered according to Act of Congress, in the year Eighteen Hundred and Fifty, BY CHARLES DAVIES, In the Clerk's Office of the District Court of the United States for the Southern District of New York. Stereotyped by RICHARD C. VALENTINE, New York. F. C. GUTIERREZ, PRINTER, Cor. John and Dutch-streets, N. Y. 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 lies at 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 yet 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 busiIt 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 three books, entitled First Lessons in Arithmetic; Arithmetic; and University Arithmeticthe latter of which is the present volume ness. The First Lessons in Arithmetic are designed for beginners. The subjects treated are divided into separat lessons, each lesson embracing one combination of numbers, or one set of combinations. The Arithmetic is designed for the use of schools and academies, and contains all that is usually taught in a course of academical instruction. The University Arithmetic is intended to answer another object. In it, the entire subject is treated as a science. The scholar is supposed to be familiar with the operations in the four ground rules, which are 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 increases from right to left according to the scale of tens, and the Denominate or Compound numbers in which it increases according to a different scale, belong in fact to the same class of numbers, and that both may be treated under a common set of rules. Hence, the rules for Notation, Addition, Subtraction, Multiplication, and Division, have been so constructed as to apply equally to all numbers. This arrangement, which the author has not seen elsewhere, is deemed an essential improvement in the science of Arithmetic. In developing the properties of numbers, from their elementary to their highest combinations, great labor has been bestowed in classification and arrangement. It has been a leading object to present the entire subject of Arithmetic as forming a series of dependent and connected propositions: so that the pupil, while acquiring useful and practical knowledge, may at the same time be introduced to those beautiful methods of exact reasoning, which science alone can teach. Great care has also been taken to demonstrate fully al the rules and to explain the reason of every process from the most simple to the most difficult. It has been thought that the Teachers of the country would like to possess a work of this kind, and that it might be studied advantageously as a text book in our advanced schools and academies. To adapt it to such a use, a large number of practical examples has been added, many of which have been selected from an English work by Keith. In the preparation of the work, another object has been kept constantly in view, viz., to adapt it to the business. wants of the country. For this purpose much pains have been bestowed in the preparation of the articles on Weights and Measures, foreign and domestic; on Banking, Bank Discount, Interest, Coins and Currency, and Exchanges. Although by law the hundred weight is estimated at 100 pounds, and consequently the quarter at 25 pounds, in the United States, yet the old hundred of 112 pounds. is still much used; and in all our intercourse with Great Britain, goods and wares are so estimated. Hence, it was thought best in this arithmetic, intended for general instruction, to retain the old standard. In fine, it has been the aim of the author to publish both a scientific and practical treatise on the subject of Arithmetic, and one which shall in some measure correspond to the higher qualifications of teachers and the improved methods of communicating instruction. Several excellent works, of an elementary character, having recently been published on Book-keeping, it has seemed best to omit, in the present edition, the article on that subject, and to supply its place by matter of a practical character. FISHKILL LANDING, JANUARY, 1850. |