the imagination in unfolding the principle and discovering the opora tions requisite for the solution. This principle is made the basis of this treatise ; viz. whenever a new combination is introduced, it is done with practical examples, proposed in such a manner as to show what it is, and as much as possible, how it is to be performed. The examples are so small that the pupil may easily reason upon them, and that there will be no difficulty in the operation itself, until the combination is well understood. In this way it is believed that the leading idea which the pupil will obtain of each combination, will be the effect which will be produced by it, rather than how to perform it, though the latter wilt be sufficiently well understood. The second part contains an analytical developement of the principles. Almost all the examples used for this purpose are practical. Care has been taken to make every principle depend as little as possible upon others. Young persons cannot well follow a course of reasoning where one principle is built upon another. Besides, a principle is always less understood by every one, in proportion as it is made to depend on others. In tracing the principles, several distinctions have been made which have not generally been made. They are principally in division of whole numbers, and in division of whole numbers by fractions, and fractions by fractions. There are some instances. also of combinations being classed together, which others have kept separate. As the purpose is to give the learner a knowledge of the principles, it is necessary to have the variety of examples under each principle as great as possible. The usual method of arrangement, according to subjects, has been on this account entirely rejected, and the arrangeinent has been made according to principles. Many different subjects come under the same principle; and different parts of the same subject frequently come under different principles. When the principles are well understood, very few subjects will require a particular rule, and if the pupil is properly introduced to them, he will understand them better without a rule than with one. Besides, he will be better prepared for the cases which occur in business, as he will be obliged to meet them there without a name. The different subjects, as they are generally arranged, often embarrass the learner. When he meets with a name with whic, he is not uainted, and rule attached to it, be is frequently at a loss, when if he saw the example without the name, he would not hesitate at all. The manner of performing examples will appear new to many, but it will be found much more agreeable to the practice of men of busi ness, and men of science generally, than those commonly found in books. This is the method of those that understand the subject. The others were invented as a substitute for understanding. The rule of three is entirely omilted. This has been considered useless in France, for some years, though it has been retained in their books. Those who understand the principles sufficiently to comprehend the nature of the rule of three, can do much better without it than with it, for when it is used, it obscures, rather than illustrates, the subject to which it is applied. The principle of the rule of three is similar to the combinations in Art. XVI. The rule of Position has been omitted. This is an artificial rule, the principle of which cannot be well understood without the aid of Algebra : and when Algebra is understood, Position is useless. Besides, all the examples which can be performed by Position, may be performed much more easily, and in a manner perfectly intelligible, without it. Tho manner in which they are performed is similar to that of Algebra, but without Algebraic notation. The principle of false position, proporly so called, is applied only to questions where there are not sufficient data to solve them directly. Powers and roots, though arithmetical operations, come more properly within the province of Algebra. There are no answers to the examples given in the book. A key is published separately for teachers, containing the answers and solutions of the most difficult examples. TABLE OF CONTENTS. (This Table equally refers to Parts I. and II.) III. Multiplication, when the multiplier is a single figure. IV. Compound numbers, factors, and multiplication, when the multi- V. Multiplication, when the multiplier is 10, 100, 1000, &c. VI. Do. when the multiplier is 20, 300, &c. Do. when the multiplier consists of any number of IX. Division, to find how many times one number is contained in X. Division. Explanation of Fractions. Their Notation. What is to be done with the remainder after division. XI. Division, when the divisor is 10, 100, &c. XII. To find what part of one number another is, or to find the ratio XIII. To change an improper fraction to a whole or mixed number. XIV. To change a whole or mixed number to an improper fraction. XV. To multiply a fraction by a whole number, by multiplying the XVI. Division, to divide a number into parts. To multiply a whole XVII. To divide a fraction by a whole number. To multiply a frac- XVIII. To multiply a fraction by dividing the denominator. Two ways to multiply, and two ways to divide, a fraction. common denominator. To reduce them to lower terms. XXI. How to find the divisors of numbers. To find the greatest com- mon divisor of two or more numbers. To reduce fractions to XXII. To find the least common multiple of two or more numbers. To reduce fractions to the least common denominator. fraction, when the purpose is to find how many times the divi- and a whole number, or of two fractions. fraction ; a part of a number being given to find the whole. XXV. Decimal Fractions. Numeration and notation of them. XXVI. Addition and Subtraction of Decimals. To change a common INDEX TO PARTICULAR SUBJECTS, 1....25 28 Compound Multiplication Page. Example 37 1....49 211 43....50 92 65..113 104 43....74 215 58....68 78 130..142 224 110..113 38 102..106 42 34....38 103 33....41 214 52....57 58 158..166 85....86 87....92 222 103..109 218 69....72 218 73....84 79 1....49 101 13....26 229 141..144 103 28....32 234 188..138 235 199..205 236 240 220 |