THE RATIONAL ARITHMETIC, IN WHICH THE SCIENCE IS FULLY DEVELOPED, THE ART CLEARLY EXPLAINED, AND BOTH COMBINED IN NUMEROUS ILLUSTRATIONS; ADAPTED TO LEARNERS OF EVERY CAPACITY. THL WHOLE ENFORCED BY A GREAT VARIETY OF INTERESTING AND PRACTICAL PROBLEMS. TO WHICH IS APPENDED, A KEY, CONTAINING THE ANSWERS TO THE PROBLEMS. BY J. S. RUSSELL, LOWELL: IS 47. Entered according to Act of Congress, in the year 1846, by J. S. RUSSELL, In the Clerk's Office of the District Court of Massachusetts. Stereotyped by GEORGEA, CURTIS; EVERY public school may be divided, in respect to the study of Arithmetic, into three classes. The first and smallest class either possess by nature, or have happily acquired, a taste, and, consequently, a talent for the study. For them there is no imperious necessity of adding another to the numerous treatises already in use; for, although they will meet with much difficulty, through indefinite and confused modes of expression and incomplete demonstrations, in arriving at the philosophy of the subject, yet, in spite of these obstacles, they will eventually comprehend the important principles of Arithmetic, and, what is remarkable, adopt the same modes of expression which have so much opposed their own progress. To this class the Rational Arithmetic, though not indispensable, will be of very essential service. Had all' learners been of this class, however, the author would have been spared the labor and expense he has devoted to this work. But there is a class of medium ability, including about one half, who may be saved incalculable labor and vexation, by using this book, while pursuing this difficult study. It is expected, however, that the third class will most truly appreciate this work." They include about one third, and consist of those unfortunate scholars whose minds act too slowly for the patience of teachers, and are too obtuse to derive much advantage from the textbooks so ill-adapted to their wants. It is for these two classes, the last in particular, that the Rational Arithmetic is prepared; to their wants it is thought to be well adapted ; and it is expected that they will hereafter assume a more just relative standing with their schoolmates ; not waiting, as heretofore, for the results of business life to prove them possessed of minds, less active indeed, yet not inferior in strength and capacity of improvement. The author knows no other written Arithmetic that is adapted to learners; they seem to him rather books of reference for those who already understand the subject, and are able to perceive the princi M305993 as ples without exp.anation. Some, indeed, have attempted to meet the wants of learners, by introducing the several subjects by a page or two of puerile questions which are seldom noticed either by teachers or scholars. Others found the principles upon the imaginary answers to be given to such questions by those acknowledged to be ignorant. Such must be a very uncertain foundation, especially so when, as sometimes happens, these leading questions are so misformed as to lead astray. Instance the following: “In 11, how much more does the 1 in the tens' place stand for than the 1 in the unit's place? In 880, how much more does the 8 in the hundreds' place stand for than the 8 in the tens' place? It is just so in all cases; therefore, A figure at the left of another stands for len times as much as it would in the place of that other figure." The simple learner will, probably, understand these questions literally, and give for answers, so far as he is able, 9 more, 720 more, &c., between which and the principle purporting to be derived' from them, there is no direct connection. Had these questions been thus, How тапу times as much .? instead of “ How much more than -? " they would have led the intelligent mind directly to the principle. In presence of the teacher to correct false answers, to sum up and enforce the conclusion, such questions properly asked are well enough; but otherwise, they are extremely vexatious and discouraging, and most scholars will pass over them without adding to their knowledge. In the Rational Arithmetic it has been the object to prepare matter for the intelligent study of the learner by himself, that in due time he may, in a well conducted recitation, exhibit with credit and pleasure, both to himself and teacher, his thorough knowledge of the lesson. Such results are far different from ordinary experience. Teachers who have desired to ground their pupils in the principles of the science, while the text-books have failed to afford the necessary instruction, have, by oral instruction, and black-board illustrations, endeavored, with only partial success, to effect this object, at present, so indispensable. Such instruction, though efficient with the more intelligent and active minds, proves insufficient for a large portion of every school. It is expected that the Rational Arithmetic will come to the relief of such teachers, enabling them with less labor to secure much happier results. The peculiarities of the Rational Arithmetic are : 1st, A philosophical arrangement, and systematic treatment of the several subjects. Multiplication and Division, being only particular cases of Addition and Subtraction, respectively, follow their heads in the natural order, in the fundamental principles ; but in fractions and compound numbers, from the greater convenience, they resume the common order gain. The subjects under the head of Percentage, being applications of the principle of Proportion, very properly follow Proportion. The study of Arithmetic being now so extensive, it is no longer necessary to place Interest nearer the beginning of the book than its proper place, to insure a knowledge of it. Indeed, all the subjects are so arranged that each is explained on principles previously taught. 2d. A complete development of the fundamental principles. Numeration, in particular, both integral and fractional, the foundation of the whole superstructure, has received especial attention. 3d. A full description and thorough explanation of the various applications of the fundamental principles. 4th. A constant regard to the abridgment of labor, by viewing numbers through their factors, and relations, canceling common factors when both multiplication and division are involved in the same process; and always operating upon fractions in a manner to secure the simplest terms in results. 5th. The giving of a reason for everything stated, and in such style that the repetition of the language will induce in the learners the understanding of the reason which it embodies. 6th. Numerous references to other parts of the book for information bearing upon the subject in hand. 7th. An exclusion of all such indefinite expressions as “ 5 times greater, seven times too large,' seven times too small," " sincreases it ten times,” “5 times too great,' ,"? « 100 times larger or smaller ;", and all such provoking expressions as “it is obvious," " “it is plain," evidently," &c., whose office is only to occupy the place of an inconvenient reason. These peculiar excellences, it is thought, warrant the title assumed for the book. It is recommended to teachers, although the author knows no written arithmetic so easily understood, that the younger pupils, previously to taking up this book, shall have well studied Colburn's First Lessons, or some other intellectual arithmetic; but when it is taken up, that they accommodate their speed to thoroughness ; that they take special notice of the numerous references to other parts of the book, where their memories may be refreshed with necessary information upon the present subject. Teachers will, of course, as far as circumstances admit, classify their pupils, assign lessons, and hear recitations in this, as in other studies. Although each is left to his own experience and tact in conducting recitations, yet we may urge the importance of securing in some way a thorough analysis of everything, the giving of a reason for each step in the solution of problems, showing its bearing upon, or tendency towards the final result. In the author's experience, it has proved well to require the pupils to bring to the recitation, not only the results, but the written process of their work; also to exhibit their skill in the solution of problems upon a black-board sufficiently ample to accommodate the whole class. The problems may be those of the ordinary lesson, or such as may be suggested on the occasion; and the recitation should be such as shall exhibit the scholar's knowledge of the principles involved in the process. The impossibility of preventing the access of the scholars to the |