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5. The principles' are arranged consecutively, and the dependence of each on those that precede it, is pointed out by references. Treated in this manner, the science of Arithmetic presents a series of principles and propositions alike harmonious and logical ; and the study of it cannot fail to exert the happiest influence in dereloping and strengthening the reasoning powers of the learner,

6. The rules are demonstrated with care, and the reasons of every opere ation fully illustrated.

7. The examples are copious and diversified; calling every principle into exercise, and making its application thoroughly understood.

8. In the arrangement of subjects, the natural order of the science has been carefully followed. Common Fractions have therefore been placed immediately after Division, for two reasons. First, they arise from division, being in fact unexecuted division. Second, in Reduction and the Compound Rules, it is often necessary to multiply and divide by fractions, to add and subtract them, also to carry for them, unless perchance the examples are constructed for the occasion and with special reference to avoiding these difficulties.

For the same reason Federal Money, which is based upon the decimal notation, is placed after Decimal Fractions; Interest, Commission, &c., after Percentage. To require a pupil to understand a rule before he is acquainted with the principles upon which it is based, is compelling him to raise a superstructure, before he is permitted to lay a foundation.

9. In preparing the Tables of Weights and Measures, no effort has been spared to ascertain those in present use in our country; and rejecting such as are obsolete, we have introduced the Standard Units adopted by the Government, together with the methods of determining and applying those standards.

10. Great labor has also been expended in preparing full and accurate Tables of Foreign Weights and Measures, and Moneys of Account, and in comparing them with those of the United States.

Such is a brief outline of the present work. In a word, it is designed to be an auxiliary to the teacher, a lucid and comprehensive text-book for the pupil, and an acceptable acquisition to the counting-room. It contains many illustrations and principles not found in other works before the public, and much is believed to be gained in the method of reasoning and analysis. No labor has been spared to render it worthy of the marked favor with which the former productions of the author have been received.

J. B. THOMSON. New York, August, 1847.

ON THE

MODE OF TEACHING ARITHMETIC.

I. QUALIFICATIONS.—The chief qualifications requisite in teaching Arithmetic, as well as other branches, are the following:

1. A thorough knowledge of the subject.
2. A love for the employment.
3. An aptitude to teach. These are indispensable to success.

II. CLASSIFICATION.--Arithmetic, like reading, grammar, &c., should be taught in classes.

1. This method saves much time, and thus enables the teacher to devote more attention to oral illustrations.

2. The action of mind upon mind, is a powerful stimulant to exertion, and cannot fail to create a zest for the study.

3. The mode of analyzing and reasoning of one scholar, will often suggest new ideas to others in the class.

4. In the classification, those should be put together who possess as nearly equal capacities and attainments as possible. If any of the class learn quicker than others, they should be allowed to take up an extra study, or be furnisheu with additional examples to solve, so that the whole class may advance together.

5. The number in a class, if practicable, should not be less than six, nor over twelve or fifteen. If the number is less, the recitation is apt to be deficient in animation; if greater, the turn to recite does not come round sufficiently often to keep up the interest.

III. APPARATUS.—The Black-board and Numerical Frame are as indispensable to the teacher, as tables and cutlery are to the house-keeper. Not a recitation passes without use for the black-board. If a principle is to be demonstrated or an operation explained, it should be done upon the black-board, so that all may see and understand it at once.

To illustrate the increase of numbers, the process of adding, subtracting, multiplying, dividing, &c., to young scholars, the Numerical Frame furnishes one of the most simple and convenient methods ever invented.

Every one who ciphers will of course have a slate. Indeed, it is desirable that every scholar in school, even to the very youngest, should be furnished with a slate, so that when their lessons are learned each one may busy himself in writing and drawing various familiar objects. Idleness in school is the parent of mischief, and employment is the best antidote against disobedience.

Geometrical diagrams and solids are also highly useful in illustrating many points in arithmetic, and no school should be without them.

· IV. RECITATIONS.—The first object in a recitation, is to secure the attention of the class. This is done chiefly by throwing life and variety into the exercise. Children loathe dullness, while animation and variety are their delight.

2. Every example should be analyzed; the "why and the wherefore' of every step in the solution should be required, till each member of the class becomes perfectly familiar with the process of reasoning and analysis.

3. To ascertain whether each pupil has the right answer, it is an excellent method to name a question, then call upon some one to give the answer, and before deciding whether it is right or wrong, ask how many in the class agree with it. The answer they give by raising their hand, will show at once how many are right. The explanation of the process may now be made.

V. OBJECTS OF THE STUDY.- :- When properly studied, two important ends are attained. ist. Discipline of mind, and the development of the reasoning powers. 20. Facility and accuracy in the application of numbers to business calculations,

VI. THOROUGHNESS.—The motto of every teacher should be thoroughness. Without it, the great ends of the study of Arithmetic are defeated.

1. In securing this object, much advantage is derived from frequent reviews.

2. Every operation should be proved. The intellectual discipline and habits of accuracy thus secured, will richly reward the student for his time and toil.

3. Not a recitation should pass without praclical exercises upon the blackboard or slates, besides the lesson assigned.

4. After the class have solved the examples under a rule, each one should be required to give an accurate account of its principles with the reason for each step, either in his own language or that of the author.

5. Mental Exercises in arithmetic are exceedingly useful in making ready and accurate arithmeticians; hence, the practice of connecting mental with written exercises, throughout the whole course, is strongly recommended.

VII. SELF-RELIANCE.—The habit of self-reliance in study, is confessedly invaluable. Its power is proverbial; I had almost said, omnipotent. 6 Where there is a will, there is a way.

1. To acquire this habit, the pupil, like a child learning to walk, must be taught to depend upon himself. Hence,

2. When assistance is required, it should be given indirectly; not by taking the slate and solving the example for him, but by explaining the meaning of the question, or illustrating the principle on which the operation depends, by supposing a more familiar case. Thus the pupil will be able to solve the question himself, and his eye will sparkle with the consciousness of victory.

3. The pupil should be encouraged to study out different solutions, and to adopt the most concise and elegant.

4. Finally, he should learn to perform examples independent of the answer. Without this attainment the pupil receives but little or no discipline from the study, and acquires no confidence in his own abilities. What though he comes to the recitation with an occasional wrong answer; it were better to solve one question understandingly and alone, than to copy a score of answers from the book. What would the study of mental arithmetic be worth, if the pupil had the answers before him? What is a young man good for in the counting-roon, who cannot perform arithmetical operations without looking to the answer ? Every one pronounces him unfit to be trusted with business calculations.

Division of Fractions, common method,

Division of Fractions by Cancelation,

Contractions in Division of Fractions, three methods,

Complex Fractions reduced to Simple ones,

SECTION VIII.

COMPOUND NUMBERS,

Tables of Compound Numbers,

The Standard for Gold and Silver Coin of the United States,

The Standard Unit of Weight of the United States,

Origin of Weights and Measures,

The Avoirdupois Pound of the United States and Great Britain,

The Standard Unit of Length of the United States,

of New York and Great Britain,

The Standard Unit of Liquid Measure of the United States,

of Dry Measure of the United States,

Method of determining whether a given year is Leap Year,

French Money, Weights and Measures,

Foreign Weights and Measures, compared with United States,

GENERAL RULE for Reduction,

Applications of Reduction,

Cubic Measure reduced to Dry, or Liquid Measure, &c.,

Longitude reduced to Time, &c.,

Compound Numbers reduced to Fractions,

Fractional Compound Numbers reduced to whole nunibers,

Compound Addition,

Compound Subtraction,

Compound Multiplication,

Compound Division,

SECTION IX.

DECIMAL FRACTIONS, their origin, &c.,

Method of reading Decimals,

Addition of Decimals,

Subtraction of Decimals,

Multiplication of Decimals,

Contractions in Multiplication of Decimals, two cases,

Division of Decimals,

Contractions in Division of Decimals,

Decimals reduced to Common Fractions,

Common Fractions reduced to Decimals,

Compound Numbers reduced to Decimals,

SECTION X.

ADDITION of Circulating Decimals,

Subtraction of Circulating Decimals,

Multiplication of Circulating Decimals,

Division of Circulating Decimals,

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