latter, the pupil reads or commits to memory the reasonings of another mind; in the former, his rules are the result of his own mental processes. He is led forward by appropriate questions; but he cannot take a single step without active intellectual employment, thus continually eliciting energy of thought, clearness of expression, and fruitfulness of invention. In the Oral Arithmetic, the lessons should be short, and the questions read slowly, till the pupils become prompt with their answers. Every section should be repeated till it is thoroughly mastered. Answers are occasionally given in the book; but this is merely to indicate the form of expression. None of the exercises will be found too difficult when taken in regular order. Obstacles can only arise from omissions. In the Written Arithmetic, one question is generally answered by the adjoining one. At other times the answer is given, unless rendered unnecessary by the mode of proof being pointed out. The exemplifications may be studied on the slate, when the teacher is unable to find time to use the black-board; but such a course is by no means to be recommended. A few of the illustrations may appear obscure when read unconnected with the computations. But all such obscurities will vanish when they are read in their proper connection. The paragraphs within brackets, [ ], are intended chiefly for the teacher. In p. 199, 1. 16 from the bottom, a question will be found leading to the formation of a formal rule by the pupil. This, or a similar question, can be repeated wherever the teacher may consider such formulas of any importance. But, where the pupil has acquired clear ideas of the principles of Arithmetic, which he cannot fail to do if he studies this book properly, formal rules will rarely, if ever, be found necessary. See p. 203 and 308—14, for another method of forming rules. When the pupil is at a loss at a computation, the teacher should neither work it out for him, nor directly instruct him how to proceed. He should merely ask one or more leading questions, or refer him to the proper passage in the book, in order to elicit thought, and lead the pupil to rely on his own resources. See p. viii. for particular instructions as to the method of using the book. CONTENTS. SECTION IV. -Increase and Decrease by a small number, causing a change in the ty, or tens, . SECTION V.-Increase and Decrease by a small number, causing a change in the tens and the hundreds, SECTION VI. — Increase and Decrease by larger numbers, the units SECTION IX. -Increase and Decrease by large numbers, the tens causing a change in the hundreds, Increase and Decrease by large numbers, the units INCREASE AND DECREASE OF FRACTIONS, OR BROKEN NUMBERS. Fractions of numbers exceeding unity, continued, SECTION VI.-Practical Questions, Addition and Subtraction of common fractions, . Contracted Addition and Subtraction of common frac- THE SHORTENED PROCESSES OF INCREASE AND DECREASE OF INTEGERS AND THE SHORTENED PROCESSES OF INCREASE AND DECREASE APPLIED TO COM- II. Addition and Subtraction of Common Fractions, III. Multiplication and Division of Common Fractions, Rapid and concise methods of computing with Common Fractions, .213 IV. Involution and Evolution of Common Fractions, Practical Exercises on Common Fractional Quantities, . 252 • 254 METHOD OF USING THIS BOOK. BEGINNERS should commence by daily practice on the Improved Numeral Frame, described in p. 11, followed by the exercises in Oral Arithmetic, Chap. I., pp. 15–66. When the learner, or class, has become familiar with a few sections in this chapter, he may commence the study of the first chapter in Written Arithmetic, pp. 108-124. The oral and written Exercises should now proceed simultaneously; the former clearing the way and facilitating the operations in the latter. Chaps. II. and III. Oral Arithmetic, should be fully mastered before the pupil commences Chap. III. Written Arithmetic. Those who have already made some progress in Arithmetic should pursue pretty much the same course, omitting, or passing rapidly over such parts as may seem familiar to them, if any such there be. It is believed, however, that this will seldom be found to be the case. See the Preface for the general OBJECT of the work. The following paragraphs on "Exchange” have been omitted by mistake in their proper place at p. 248. STERLING may be changed to Canada money by adding to its amount; Why? [See Table of Provincial Currencies, p. 227.] It may be changed to New England by adding ; Why? To New York by adding 7; Why? To Pennsylvania by adding ; Why? To South Carolina by adding '7; Why ? CANADA may be changed to New England money by adding ; to New York by adding g; to Pennsylvania by adding §. NEW ENGLAND may be changed to New York by adding }; to Pennsylvania, 15 4 PENNSYLVANIA may be changed to New York by adding SOUTH CAROLINA may be changed to Canada by adding ; to New England by adding ; to New York by adding §; to Pennsylvania by adding 17. Every one of these operations may be reversed by subtracting instead of adding the proportionate part, after changing the respective fraction by adding the numerator to the denominator for a new denominator, and allowing the numerator to remain as before. Thus in changing Canada to Sterling money, the fraction must be changed to; while New York to Sterling requires to be changed to 7. Why is this so? The 7 reason will be readily discovered by attentively operating with a fraction that has 1 for numerator, and then enlarging it to 2, 3, &c. The same principles are applicable to Foreign Exchange. |