RULES OF THREE, SINGLE AND DOUBLE, DIRECT AND INVERSE; CURRENCIES; INTEREST. AND ALL PROPORTIONAL QUESTIONS IN ONE RULE APPLICABLE TO THE WHOLE THE PROCESS GREATLY SIMPLIFIED AND ABRIDGED. BY CHARLES G. BURNHAM, A. M. KEENE, N. H.: PUBLISHED BY S. & G. S. WOODWARD HARVARD Entered according to Act of Congress, in the year 1857, by In the Clerk's Office of the District Court of the District of Massachusetts. PREFACE. HE who writes a book in this age merely for the sake of being a bookmaker, will find that he has written for other times than these; and his fame will be like one of those "second sights," Laving 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, and in its time, is the plan. The simple rules are presented in their order singly; then in contrast; then a review of the whole, to exercise the judgment of the scholar. Fractions are introduced as the result of division, or rather as division implied. They are made to occupy the same position, and are illustrated and solved the same as whole numbers. The same numbers are again written in the fractional form, and the scholar is enabled to perceive, at a single view, that a change of position, and of names, is a matter of convenience and not of necessity. În the ordinary mode of presenting fractions, the idea is not precluded from the mind of the scholar, that new positions and new names do not necessarily introduce new principles. The result is, that he perceives no connection between the present and the past, and consequently the subject is ever new, and new difficulties are constantly arising. A new form of notation, and new names being introduced, it is in vain to insist that no new principle is employed, so long as the subject is but imperfectly illustrated, and the scholar does not perceive that the change is not a matter of necessity. is one thing to gain the assent of the pupil to a truth, and it is often quite another to give him a practical understanding of it. It It is a fact too little realized, that much time is consumed in going over ground, from which no practical knowledge is gained. Not that the studies themselves are not practical, but they are not pursued in a practical manner. The scholar may be often informed that a fraction is the result of division; that the fractional form of writing numbers is division implied; and that numerator is the same as dividend, and denominator is the same as divisor; and yet difficulties will arise which did not occur in whole numbers. Whereas, a practical knowledge of this fact would enable him to solve most questions, in fractions, with the same facility as in whole numbers; nor would he find any necessity for some half dozen rules, which he is usually required to commit to memory. When the simple rules are thoroughly understood, the pupil may be introduced to the subject of fractions, in a manner similar to the following, at the blackboard. If we divide 2 by 2, the quotient is a unit or 1, 2|2=1, for the dividend is just equal to the divisor. Were we required to divide 1 by 2, we should meet with a difficulty, for the dividend is less than the divisor, and consequently will not contain it; we must therefore employ a new form of notation, 2|1=. We write the divisor under the dividen 1, and give a new name to the expression; we call it a fraction, which means a part of a thing. The quotient usually shows how many times the dividend contains the divisor. If the quotient is 2, the dividend contains the divisor twice; if 3, three times. But here the quotient is a fraction, less than a unit, or 1, which shows that the dividend is only a part of the divisor. But what part? The same part the quotient is of a unit. But what part is the quotient of a unit? It will now be convenient to introduce new names, in order to value the fraction. You perceive, that the number which we employed as divisor, we have written under the line, and the number employed as divi dend, is above the line. If our divisor be 2, our quotient is one-half of the dividend; if our divisor be 3, the quotient is one-third of the dividend. Thus it is plain, that in whole numbers, the divisor gives name to the quotient. The same is true when we imply division and write the numbers in the form of a fraction. Our divisor in this example is 2. and our quotient is one-half of the dividend: it is also one-half of a unit. The unit is divided into two parts; our quotient is now denominated; we therefore call the figure below the line, denominator, or namer, because it gives name to the parts into which the unit is divided. Thus we have our fraction named, or denominated; but what is its value? It is halves, but how many halves does it contain? Evidently one, which the figure above the line shows. We have now the fraction denominated or named, and numbered. Its denomination is halves, and their number is one. Making use of the figure above and below the line in one expression, we call the fraction one half, or one-half. Thus you perceive that numerator is the same as dividend, and denominator the same as divisor. And, as in division multiplying the dividend increased the quotient, so in fractions, multiplying numerator increases the value of the fraction. Thus: Let the scholar write numbers in this manner, side by side, and be exercised, as in division, by multiplying dividend and divisor, numerator and denominator, employing the language of division and the language of fractions, until he is practically familiar with the fact that the principle employed in fractions and whole numbers is the same. Whenever new names are introduced, and new positions employed, let the different forms be written side by side, and extra exercises be given, until the scholar clearly perceives the unity of the principle. (See example under Art. 147.) In Decimal Fractions, also, the points in which they are like whole numbers and common fractions, and points in which they differ, are distinctly brought out as the scholar proceeds, and then, at the close, those points are presented in one general view. In Proportion, new names and new positions are again employed. Let the same pains be taken to contrast the new positions with the former, and to explain the new terms introduced. TO TEACHERS. It cannot be expected that a School Arithmetic, limited in size as it must be, should exhaust its subjects, or give all those illustrations which might be both interesting and useful. The most it can do upon any one subject is to give a single illustration of a principle, a formula of a particular mode of teaching. And that text-book is the best, which by its connection of thought and subjects, and illustration of principle, interests both teacher and scholar, and incites the teacher to invent new modes for himself. Teachers are here presented with an Arithmetic which is the result of much experience in teaching and effort at improvement. It has been the purpose and aim of the author to prepare a work which should accord with the spirit of the age, and be adapted to the schoolroom. It is not expected, nor is it desirable, that the teacher should be confined to the forms laid down in the book. They are designed simply to open the subject-to serve as hints to something betThe peculiar mode of stating questions for the convenience of cancelling and for illustrating fractions as whole numbers, teachers can ter. |