PRE FACE. The book here offered to Schools and Academies, had its origin in the urgent want the author has found, in the case of his own pupils, of a higher work on Mental Arithmetic. Such a work, he has thought, should be constructed with reference to several important objects. It should habituate the pupil to perform, with ease and readiness, mental operations upon somewhat large numbers. It should present these operations in their natural form, freed from the inverted and mechanical methods which belong of necessity to operations in written Arithmetic. It should train the student to such a power in apprehending the relations of numbers, as shall give him an insight into the grounds of the rules of Arithmetic; and, consequently, shall release him from dependence on those rules; and it should free him from the liability to those wide mistakes often made in written Arithmetic, which appear so absurd, and are yet too frequent to excite the teacher's surprise. A higher training in Mental Arithmetic would also, it is believed, prepare the members of our schools, when they should leave their studies and engage in the active pursuits of life, to solve mentally, and with ease and delight, a large share of those questions, of business or curiosity, for which a process of ciphering is ordinarily thought indispensable. The study of Arithmetic in the schools of this country received its best impulse, unquestionably, in the publication of “ Colburn's First Lessons.” So completely has this little book performed the work within its prescribed sphere, that there is little reason to desire a change in that particular, or to expect that the work will, for the present, be superseded. Whoever would now write a book of First Lessons in Arithmetic, must, it is believed, if he would write a good one, walk the most of his way in the steps of one, at least, who has gone before him. The “ Advanced Lessons” are designed to continue and extend the course of discipline in numbers, which is begun in the elementary book above named. Consequently it requires, for its successful study, an acquaintance with the elements, as taught in that work, or in some other occupying essentially the same ground. In all the mental calculations in large sums, it will be found a uniform characteristic of this work to begin with the highest order of numbers in the sum, - hundreds before tens, tens before units. In this way, the numbers are presented in the same order in which they are presented in the common usage of our language. In most of the operations of written Arithmetic, however, the smallest number is taken first; and thus a method is pursued, the reverse of what the genius of our language would naturally suggest. Another advantage of taking the highest numbers first, in Mental Arithmetic is, that we thus obtain a large approximation to the final answer, at the first step. When the first step, however, as in written addition, or multiplication, furnishes only the units of the answer, leaving the hundreds or thousands still unknown, only a minute fraction of the answer is at first obtained. It is too plain to require proof, that that method will be most interesting and gratifying to the mind, which secures the largest portion of the answer at the first step. Another advantage of the method here used, is found in the fact, that we naturally make the higher order the standard, and the lower order takes its value in the mind from a comparison with the higher, as a certain part of it. Thus 150 is apprehended by the mind, as one hundred and half a hundred. This is not, indeed, the method of acquiring the idea of large numbers, but the method of combining them after the idea has been acquired; consequently, it is the legitimate method of instruction, just as soon as the pupil is qualified to enter on the study of such combinations. If, now, we obtain the number of the highest order first, we have a standard, under which all the succeeding orders naturally fall, and from a comparison with which they successively take their value. If we begin with units, however, and work upward through the higher orders, we obtain no standard; we must hold the successive numbers in suspense, until the last term shall furnish the nucleus for the group, — the standard under which all the lower orders shall take their rank. It is on the basis of these facts, which are only indications of the laws of the mind, that, throughout the Mental part of this Arithmetic, the author has in all operations, taken the highest order of numbers first. The increased interest which the persevering use of this method will awaken in the minds of pupils, will be, to teachers, a better commendation of its correctness, than any more extended mental analysis. There are other features of the Advanced Lessons which are, perhaps, sufficiently distinctive to justify their mention here ; but as the truest test of a school book is its use in the school room, the work is referred to that ordeal. The Second Part contains examples in Written Arithmetic on all the most important rules. They are designed to be sufficiently numerous to lead the student to ready and accurate practice in ciphering. In this part the author has aimed to interest the scholar by furnishing him with natural and reasonable questions, and to aid both teacher and scholar by arranging them progressively. The rules and explanations will, probably, be found sufficient, after a thorough mastery of the First Part. It is not necessary that the pupil complete the First Part before beginning the Second. He may carry on both Parts at the same time; but, under each particular head, the mental part should be thoroughly mastered before the written examples are begun. The answers to the questions in the Second Part are given in a separate work. This course has seemed to the author, on the whole, the best, notwithstanding some incidental disadvantages that may arise from it. It will enable the teacher to oversee a much larger amount of work in Arithmetic, than he could otherwise attend to. And it is believed there will not be much difficulty, if the teacher pursues a right course, in awakening in the members of a school a spirit of honor and uprightness, that will make them scorn to resort to any dishonest use of a key. To aid in awakening a higher interest and zeal in this branch of study, the author will offer a few suggestions. Let the key be used as little as the teacher's necessities will permit. Let original questions be proposed by the teacher in connexion with' every Section. Each member of the class should be encouraged to propose original questions to be solved by the class. It will often be useful, especially in a review, to alter some one figure in the conditions of each question. This often produces a happy excitement, and gives quite a new zest to the study. DUMMER ACADEMY, April 18, 1846, CONTENTS. PART FIRST. I. Multiplication of Tens and Units, II. Multiplication of Tens and Units. — Complement, . 16 VI. Federal Money. - Sterling Money. — Dry Measure. -- Avoirdupois Weight.- Troy Weight.— Apothe- caries' Weight. — Cloth Measure.—Wine Measure. Beer Measure. Measure of the Circle, VIII. Multiplication and Division of Fractions. — To Find IX. Multiplication of Fractions by Fractions. Division of Fractions by Fractions. Addition of Frac- tions. — To find a Common Denominator, XII. Decimal Fractions. - Addition and Subtraction of 71 XIII. Reduction of Vulgar Fractions to Decimals, XIV. Interest. Banking Discount. Loss and Gain. - 75 - Per Centage, |