ELEMENTS OF ARITHMETIC; FOR SCHOOLS AND ACADEMIES. IN WHICH DECIMAL AND INTEGRAL ARITHMETIC ARE COMBINED, AND TAUGHT INDUCTIVELY, ON THE SYSTEM OF PESTALOZZI. PART SECOND. BY PLINY E. CHASE. PHILADELPHIA: URIAH HUNT AND SON. due T 118,44, 277 Entered, according to the Act of Congress, in the year 1844, by PLINY E. CHASE, in the clerk's office of the District Court of the United States in and for the Eastern District of Pennsylvania. PREFACE. The improvements which have been made within a few years, in the mode of teaching Arithmetic, and the farther improvements which may result from a philosophical arrangement, and a more satisfactory explanation of some principles of the science, were the inducements which led to the compilation of the present series. In the preparation of the original manuscript, I was guided by a knowledge of the wants of my own pupils; and after I had been advised to publish the result of my labours, I studied to acquaint myself with the best European and American treatises on the subject, and to derive from experienced practical teachers* such information as would assist the pupil, by removing the most formidable obstacles from his path. Particular care has been taken to illustrate all the principles of Fractions, and to simplify the general arrangement, by treating in connection all subjects of a similar nature. The work has in this manner been greatly condensed, and an opportunity afforded for giv. ing full explanations of every difficult point, and embracing within a small compass, a greater variety of subjects than is contained in any of our present treatises. The most important ORIGINAL feature of the work, is the union of decimals with integers in the simple rules,-a mode of instruction that is rapidly gaining ground, and is sanctioned by the practice of many most successful teachers. Its practicability having been fully tested, the advantage derived from dispensing with many of the old rules, which are entirely superfluous, and tend to retard the progress of the pupil, will be readily appreciated. The Rule of Three has lost much of its ancient reputation, and some of its applications have been very properly dismissed from our modern text-books. Still there are many examples which involve the theory of Proportion, and to their solution the rule given on page 88 (which is essentially the same as the one employed in the best schools of England and the Continent) can be applied with great facility. The Rule for multiplying in a single line will interest by its novelty, and afford an excellent exercise for giving expertness in adding columns of two figures. It is not recommended for universal adop * My obligations are particularly due to Samuel Alsop, Principal of Friends' Select School, and WILLIAM J. Lewis, Teacher of the Mathematical School in Fourth street, (5) tion, although scholars of a decided mathematical taste will soon learn to apply it,-particularly where each factor consists of twenty or more figures,—with less liability to error than the ordinary rule. The Algebraic symbols + and —, with a few others, have long been employed in our Arithmetics. Can any valid objection be made to the like use of letters, to represent the unknown quantity ? It is certainly much more convenient to say “725," than “7 times the 5th power of a certain number.” The pupil will readily understand the application of letters, if he is told to work with them as he would with the answer to prove its correctness, and he will congratulate himself on the acquisition of a key to the many mysteries of Analysis. Holdred's General Rule for the Extraction of Roots is universally admitted to be the best that has ever been proposed, and has there. fore been substituted for the common rule, which is equally Algebraical, and much more tedious in its application. The chapters on Divisibility of Numbers, and Numerical Approx. imations, will be found very useful to every one who desires to be. come thoroughly versed in the principles of Arithmetic. A knowledge of the theory of prime numbers (in connection with the table, which has been entirely re-calculated, and is now believed to be perfectly accurate) is of great advantage to the mathematician, and every student will find it valuable in reducing fractions to their lowest terms, and in cancelling the factors common to long multipliers and divisors. The attention of teachers is likewise invited to the rule for divid. ing by 9's, the remarks on Transposition, the contractions at the close of the chapter on Practice, the Explanation of the Square Root, and the collection of valuable Tables at the close of the present vol. ume. 1 The examples are of such a character as to serve the purpose of a constant review, as in each chapter the principles of the preceding chapters are involved, and the pupil is frequently led to apply all the rules with which he has become familiar. In all respects it has been my aim to render the work practical, and at the same time rigorously mathematical. I have sought to prepare a book at once simple, plain, concise, accurate and philosophical in its details, and it is now submitted to the criticism of an im. partial public to judge of my success. PLINY E. CHASE. Philadelphia, 1844. |