of problems involving the transactions of every-day life. In this work an effort has been made to avoid these faults, as well as to furnish the pupil with ample material, systematically graded, to give that discipline of the mind which will fit him for the practical duties of life. The work is divided into three parts; the First part contains a complete theoretical review of the subjects contained in the INTERMEDIATE, with copious test questions and problems. The Second part of the GRAMMAR SCHOOL is a continuation of the INTERMEDIATE, and contains, with the INTERMEDIATE, a thorough, short, common-school course, emphatically practical, and complete in all the ordinary transactions of the work-shop, market, store, and bank. Part Third contains a more advanced course of commercial instruction, including the Progressions, Permutations, Duodecimals, etc., and the principal problems in Philosophy and Mechanics, giving all the instructions necessary in Commercial Colleges, Academies, and High-schools. The Second part is bound with the INTERMEDIATE in one volume, and · forms what its name indicates, the PRACTICAL. The author wishes to acknowledge his indebtedness to the numerous teachers who have shown an interest in this series by kindly offering valuable suggestions, as well as to R. L. Delisser, Esq., for his method of averaging accounts by interest, which he has kindly given us permission to use. BROOKLYN, N. Y., Sept., 1868. INTRODUCTION. In presenting this work to his fellow teachers, the author deems it proper to exhibit its peculiarities more fully than he can do in a preface for the general reader. ANALYSIS. This work is intended to continue the subject of arithmetic as left in the Intermediate; and, also, to make it in some measure complete for those who wish a theoretical review of the subjects treated in that volume. All the tables, principles, analyses, formulas, &c., given in the Intermediate are, therefore, reproduced in this book, with test problems and questions, by which the formation of problems as well as their true analysis is clearly exbibited. Every complex, concrete, or commercial problem can be separated into elementary questions, and each question involves one of six arithmetical formulas. Using the symbols x and y, we have: I. If some oranges cost x cts., and some apples cost y cts., both will cost the sum of these quantities, which is x+Y cts. II. If the oranges cost x cts., and the apples cost Y cts., the oranges will cost as many more cents than the apples as the difference of these quantities, which is X-Y cts. III. If one orange cost x ets., Y oranges will cost y times x cls., which is XXY cts. IV. If x cts. are equally divided among y boys, each boy will receive one Y part of x cts., which is x÷Y cts. V. If one orange costs Ÿ cts., as many oranges can be bought for x cts. as Y cts, are contained times in x cts., which is x÷Y times. Y VI. If a boy have x cts. and he give away Y cts., he will give away the part of his money. Substituting 18 as the value of x, and 9 as the value of y, we have the numerical values of each respectively, as follows: 27 cts., 9 cts., 162 cts., 2 cts., 2 oranges and or of his money. By substituting other numerical values and denominations a large number of examples may be formed, and, by combination, the ingenious teacher may find pleasant, and the pupil, profitable employment. While undue prominence should not be given to this method, it is important that in the analysis of problems the pupil be continually required to . point out the more important of these elementary questions, and their appropriate formulas. By this means, he will become self-reliant, unaccustomed to flee for refuge in every difficulty to some RULE; but will depend upon the exercise of his reason for a solution. ARRANGEMENT.-It has been objected that, in a logical order of subjects, fractions should precede denominate numbers, and U. S. Currency should be placed with decimals, where it logically belongs. The author would say, in answer, that since no scale can be added or subtracted without reduction, this logic would require reduction to be taught before addition, and the complete decimal scale before fractions or denominate numbers. He would not follow out this absurdity, but counsel the natural as the logical order. All will agree that a child comprehends the simple before the complex. What, then, is the simplest subject in arithmetic? Certainly it is the integral portion of the decimal scale. The reduction of this is so simple that it is omitted, and the fundamental rules follow. None will dispute that the integral portion of the denominate scale is more simple than the fractional scale, and hence should precede it. An objection is sometimes raised that fractions are found in denominate numbers; but, certainly, those few fractions can be more easily explained than the whole subject of fractions, and the objection is, therefore, not valid. This arrangement of subjects has the further advantage, that, after the scholar has passed through the fundamental rules, he is made to apply his knowledge to business transactions in the currency which he is daily using, and is thus made to feel that arithmetic really means something in life. He then gets a thorough knowledge of the application of the tables of denominate numbers with especial reference to their use in purchases and sales. The pupil is better fitted to leave school with a thorough knowledge of these subjects, and the discipline which the acquisition will give him, than with a knowledge of the simple operations of fractions and decimals combined; for the latter he will soon forget, while the former will constitute a capital at compound interest. OMISSIONS AND ADDITIONS.-A number of subjects which are usually introduced early in the course are omitted in this treatise until Part third, for the purpose of giving place to those which are more practical. A number of subjects are added to Part third which are usually omitted, although practically of great importance. REVIEW. The method of review in this series is such that a pupil entering at almost any stage of progress will be required to review all the subjects over which the class have passed, before finishing the book, thus avoiding the necessity of turning the class back on account of a few of its members. The test questions for examination are a new and it is hoped a valuable feature of this series. |