A NEW SYSTEM OF ARITHMETIC, ON AN IMPROVED PLAN: EMBRACING THE RULES OF THREE, SINGLE AND DOUBLE, DIRECT AND INVERSE; AND ALL PROPORTIONAL QUESTIONS IN ONE RULE APPLICABLE TO THE WHOLE THE PROCESS GREATLY SIMPLIFIED AND ABRIDGED. BY CHARLES G. BURNHAM, A. M. NEW YORK: D. APPLETON & COMPANY, 200 BROADWAY. PHILADELPHIA: GEO. S. APPLETON, 164 CHESNUT STREET. MDCCC L. Entered according to Act of Congress, in the year 1849, By D. APPLETON & COMPANY, In the Clerk's Office of the District Court of the United States for the Southern District of New York. 1 PREFACE. He who writes a book in this age merely for the sake of being a book. maker, will find that he has written for other times than these; and his fame will be like one of those "second sights," having existence only in the mind of him who sees it. Every invention, every thing new, every book, from the child's primer to the most profoundly scientific text-book, must be tested by a comparison with others, professing, each and all, to be the best extant. Nor will any production gain for its author that for which he labored, unless it finally proves to be what it professes. Improvement is the "charmed word" of the age. It rings hourly in the ear of the multitude. The strong wind bears it onward, and the gentle zephyr wafts its echo. He who has already written his name far above his competitors, now seeks to outdo himself; and the tyro fancies, that he can begin where the best have ended, and run his race alone. The author of this series of Arithmetics cherishes none of these fancies, having already received satisfactory compensation for all his toil, in imparting, from time to time, to those he has had the pleasure to instruct, the improvements here embodied. But should it be found, when the decisive test has been applied, that he has said some things not before said, which may be of benefit to teachers, and the cause of education generally, his pleasure will not be less, because he had ventured to indulge some slight anticipation of the fact. If the experience of some twenty-five years in teaching has not failed to discover to him the real wants of our schools, then it will be found that his series of Arithmetics is adapted to meet those wants, and is in some measure suited to the spirit of the age in which we live. The Cancelling Arithmetic, published in 1837, was the first work known to the author, which, to any considerable extent illustrated, and practically applied the principle of cancelling. Although it is true that the principle is coextensive with the science of numbers, for no question in Simple Division can be solved without employing it, still Division was not explained as embodying the whole of it, nor was the principle so applied and illustrated as to simplify Division. The mode of writing numbers for the convenience of cancelling, in connection with the ordinary mode, affords a variety of illustrations interesting and useful, both to teachers and scholars. The application of the cancelling principle is not, however, the only peculiar characteristic of this work. It aims throughout, by the connection of its subjects, and illustration of principle, to impress upon the mind of the scholar the truth, that he will never discover nor need a new principle beyond the simple rules. Hence the first object is to make the scholar thoroughly acquainted with those rules. One thing at a time, |