THE PROGRESSIVE HIGHER ARITHMETIC, FOR SCHOOLS, ACADEMIES, AND MERCANTILE COLLEGES. COMBINING THE ANALYTIC AND SYNTHETIC METHODS; AND FORMING A COMPLETE TREATISE ON ARITHMETICAL BUSINESS APPLICATIONS. BY HORATIO N. ROBINSON, LL. D., AUTHOR OF WORKS ON ALGEBRA, GEOMETRY AND TRIGONOMETRY, SURVEYING AND NEW YORK: IVISON, PHINNEY CO., 48 AND 50 WALKER ST. BOSTON BROWN & TAGGARD. PHILADELPHIA: SOWER, BARNES & CO., AND Robinson's Complete Mathematical Course. ROBINSON'S SYSTEM OF MATHEMATICS, Recently revised and enlarged, is now the most extensive, complete, practical and scientific Mathematical Series published in this country. 1. Robinson's Progressive Primary Arithmetic. Illustrated. $0 15 2. Robinson's Progressive Intellectual Arithmetic, for ad- 0 25 3. Robinson's Progressive Practical Arithmetic: a complete 056 4. Key to Robinson's Progressive Practical Arithmetic. 0 50 0 75 6. Key to Robinson's Progressive Higher Arithmetic. . 0 75 7. Robinson's New Elementary Algebra: a clear and simple 8. Key to Robinson's New Elementary Algebra. 0 75 075 9. Robinson's University Algebra: a full and complete Trea- 10. Key to Robinson's University Algebra; separate. 13. Robinson's Analytical Geometry and Conic Sections; made clear and comprehensive to common minds. 14. Robinson's Differential and Integral Calculus; a full and complete Treatise. 1 50 15. Robinson's Elementary Astronomy; designed to teach the first principles of this Science. 075 16. Robinson's University Astronomy; for advanced classes in Academies and Colleges. 17. Robinson's Concise Mathematical Operations: a book of reference for the Teacher, embracing the gems of Mathematical Science. 18 Key to Robinson's University Algebra, Geometry, Surveying, and Calculus; in 1 vol. Entered, according to Act of Congress, in the year 1860, by D. W FISH AND J. H. FRENCH, 1 50 in the Clerk's Office of the District Court of the United States for the Northern District of New York. JOHN FAGAN, STEREOTYPER, PHILADELPHIA. PREFACE. THIS work is intended to complete a well graded and progressive series of Arithmetics, and to furnish to advanced students a more full and comprehensive text-book on the Science of Numbers than has before been published; a work that shall embrace those subjects necessary to give the pupil a thoroughly practical and scientific arithmetical education, either for the farm, the workshop, or a profession, or for the more difficult operations of the countingroom and of mercantile and commercial life. There are two general methods of presenting the elements of arithmetical science, the Synthetic and the Analytic. Comparison enters into every operation, from the simplest combination of numbers to the most complicated problems in the Higher Mathematics. Analysis first generalizes a subject and then develops the particulars of which it consists; Synthesis first presents particulars, from which, by easy and progressive steps, the pupil is led to a general and comprehensive view of the subject. Analysis separates truths and properties into their elements or first principles; Synthesis constructs general principles from particular cases. Analysis appeals more to the reason, and cultivates the desire to search for first principles, and to understand the reason for every process rather than to know the rule. Hence, the leading method in an elementary course of instruction should be the Synthetic, while in an advanced course it should be the Analytic. The following characteristics of a first class text-book will be obvious to all who examine this work: the typogra phy and mechanical execution; the philosophical and scientific arrangement of the subjects; clear and concise definitions; full and rigid analyses; exact and comprehensive rules; brief and accurate methods of operation: the wide range of subjects and the large number and practical character of the examples-in a word, SCIENTIFIC ACCURACY combined with PRACTICAL UTILITY, throughout the entire work. Much labor and attention have been devoted to obtaining correct and adequate information pertaining to mercantile and commercial transactions, and the Government Standard units of measures, weights, and money. The counting-room, the bank, the insurance and broker's office, the navy and ship-yard, the manufactory, the wharves, the custom-house, and the mint, have all been visited, and the most reliable statistics and the latest statutes have been consulted, for the purpose of securing entire accuracy in those parts of this work which relate to these subjects and departments. As the result of this thorough investigation, many statements found in most other arithmetics of a similar grade will not agree with the facts presented in this work, and simply because the statements in these other books have been copied from older works, while laws and customs have undergone great changes since the older works were written. New material and new methods will be found in the several subjects throughout the entire work. Considerable prominence has been given to Percentage and its numerous applications, especially to Stocks, Insurance, Interest, Averaging Accounts, Domestic and Foreign Exchange, and several other subjects necessary to qualify students to become good accountants or commercial business men. And while this work may embrace many subjects not necessary to the course usually prescribed in Mercantile and Commercial Colleges, yet those subjects requisite to make good accountants, and which have been taught orally in that class of institutions from want of a suitable text-book, are fully discussed and practically applied in this work; and it is therefore believed to be better adapted to the wants of Mercantile Colleges than any similar work yet published. And while it is due, it is also proper here to state that J. C. Porter, A. M., an experienced and successful teacher of Mathematics in this State, and formerly professor of Commercial Arithmetic, in Iron City Commercial College, Pittsburgh, Penn., has rendered valuable aid in the preparation of the above-named subjects, and of other portions of the work. He is likewise the author of the Factor Table on pages 72 and 73, and of the new and valuable improvement in the method of Cube Root. Teachers entertain various views relative to having the answers to problems and examples inserted in a text-book. Some desire the answers placed immediately after the examples; others wish them placed together in the back part of the book; and still others desire them omitted altogether. All these methods have their advantages and their disadvantages. If all the answers are given, there is danger that the pupil will become careless, and not depend enough upon the accuracy of his own computations. Hence he is liable to neglect the cultivation of those habits of patient investigation and self-reliance which would result from his being obliged to test the truth and accuracy of his own processes by proof,—the only test he will have to depend upon in all the computations in real business transactions in after life. Besides, the work of proving the correctness of a result is often of quite as much value to the pupil as the work of |