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and in its time, is the plan. The simple rules are presented in their order singly; then in contrast; then a review of the whole, to exercise the judgment of the scholar. Fractions are introduced as the result of division, or rather as division implied. They are made to occupy the same position, and are illustrated and solved the same as whole numbers. The same numbers are again written in the fractional form, and the scholar is enabled to perceive, at a single view, that a change of position, and of names, is a matter of convenience and not of necessity. În the ordinary mode of presenting fractions, the idea is not precluded from the mind of the scholar, that new positions and new names do not necessarily introduce new principles. The result is, that he perceives no connection between the present and the past, and consequently the subject is ever new, and new difficulties are constantly arising. A new form of notation, and new names being introduced, it is in vain to insist that no new principle is employed, so long as the subject is but imperfectly illustrated, and the scholar does not perceive that the change is not a matter of necessity. It is one thing to gain the assent of the pupil to a truth, and it is often quite another to give him a practical understanding of it.

It is a fact too little realized, that much time is consumed in going over ground, from which no practical knowledge is gained. Not that the studies themselves are not practical, but they are not pursued in a practical manner. The scholar may be often informed that a fraction is the result of division ; that the fractional form of writing numbers is division implied, and that numerator is the same as dividend, and denominator is the same as divisor ; and yet difficulties will arise which did not occur in whole numbers. Whereas, a practical knowledge of this fact would enable him to solve most questions, in fractions, with the same facility as in whole numbers; nor would he find any necessity for some half dozen rules, which he is usually required to commit to memory.

When the simple rules are thoroughly understood, the pupil may be introduced to the subject of fractions, in a manner similar to the following, at the blackboard. we divide 2 by 2, the quotient is a unit or 1, 212=1, for the dividend is just equal to the divisor. Were we required to divide 1 by 2, we should meet with a difficulty, for the dividend is less than the divisor, and consequently will not contain it; we must therefore employ a new form of notation, 2|1=. We write the divisor under the dividend, and give a new name to the expression ; we call it a fraction, which means a part of a thing. The quotient usually shows how many times the dividend contains the divisor. If the quotient is 2, the dividend contains the divisor twice; if 3, three times. But here the quotient is a fraction, less than a unit, or 1, which shows that the dividend is only a part of the divisor. But what part? The same part the quotient is of a unit. But what part is the quotient of a unit ?

It will now be convenient to introduce new names, in order to value the fraction. You perceive, that the number which we employed as divisor, we have written under the line, and the number employed as dividend, is above the line. If our divisor be 2, our quotient is one-half of the dividend ; if our divisor be 3, the quotient is one-third of the dividend. Thus it is plain, that in whole numbers, the divisor gives name to the quotient. The same is true when we imply division and write the numbers in the form of a fraction. Our divisor in this example is 2, and

2

our quotient is one-half of the dividend: it is also one-half of a unit. The unit is divided into two parts; our quotient is now denominated; we therefore call the figure below the line, denominator, or namer, because it gives name to the parts into which the unit is divided. Thus we have our fraction named, or denominated; but what is its value? It is halves, but how many halves does it contain? Evidently one, which the figure above the line shows. We have now the fraction denominated or named, and numbered. Its denomination is halves, and their number is one. Making use of the figure above and below the line in one expression, we call the fraction one half, or one-half. Thus you perceive that numerator is the same as dividend, and denominator the same as divisor. And, as in division multiplying the dividend increased the quotient, so in fractions, multiplying numerator increases the value of the fraction. Thus:

21

1X2=2

-=1.

2

22=1 Let the scholar write numbers in this manner, side by side, and be exercised, as in division, by multiplying dividend and divisor, numerator and denominator, employing the language of division and the language of fractions, until he is practically familiar with the fact that the principle employed in fractions and whole numbers is the same.

Whenever new names are introduced, and new positions employed, let the different forms be written side by side, and extra exercises be given, until the scholar clearly perceives the unity of the principle. (See example under Art. 147.) In Decimal Fractions, also, the points in which they are like whole numbers and common fractions, and points in which they differ, are distinctly brought out as the scholar proceeds, and then, at the close, those points are presented in one general view. In Proportion, new names and new positions are again employed. Let the same pains be taken to contrast the new positions with the former, and to explain the new terms introduced.

TO TEACHERS.

It cannot be expected that a School Arithmetic, limited in size as it must be, should exhaust its subjects, or give all those illustrations which might be both interesting and useful

. The most it can do upon any one subject is to give a single illustration of a principle, a formula of a particular mode of teaching. And that text-book is the best, which by its connection of thought and subjects, and illustration of principle, interests both teacher and scholar, and incites the teacher to invent new modes for himself. Teachers are here presented with an Arithmetic which is the result of much experience in teaching and effort at improvement.

It has been the purpose and aim of the author to prepare a work which should accord with the spirit of the age, and be adapted to the schoolroom. It is not expected, nor is it desirable, that the teacher should be confined to the forms laid down in the book. They are designed

simply to open the subject to serve as hints to something bet

The peculiar mode of stating questions for the convenience of cancelling and for illustrating fractions as whole numbers, teachers can

ter.

ment.

adopt, or apply the principle of cancelling to the ordinary mode of state

It will be well to employ both modes, as together they open a wider field for illustration.

It is sometimes remarked of the cancelling system, that it is good as far as it goes. The same may be said of arithmetic; for the principle is inseparable from it. It is the only principle by which any question in division can be performed. Wherever it cannot be applied, the numbers must be written in the form of a fraction. When the question involves multiplication and division, it will generally be found to be a great saving of labor, to write down all those numbers which are to be factors of the dividend and divisor, before proceeding to the operation. The eye will then detect at a glance equal factors, and they can be excluded from the operation. The teacher will bear in mind the importance of giving general illustrations of arithmetical principles, whenever it can be done, as its tendency is to enlarge the views of the pupil and to give importance to the study For example, let simple division be illustrated not only arithmetically, but on general principles. Let it be required to divide 16 by 8, and it may be done and illustrated in the following manner :

8)16=8X2--8=2 Ans. Now substitute the letter a for 8, and the letter b for 2, and read the question thus : divide ab by a.

X) db

6 Ans. Here, as before, we exclude from the dividend a factor equal to the divi

But this latter process is algebraic; hence the scholar's views are extended, and he perceives at once, and for the first time, the connection between arithmetic and algebra. Formulas are also given to aid the less experienced teacher, and also to bring out more prominently arithmetical principles.

sor.

MANNER OF RECITATION.

Promptness and dispatch are characteristics of our times, and young men must be educated in reference to them. There is no place, perhaps, better calculated to train a scholar to think and act with precision and energy, than at the blackboard. When a scholar is called out from his class to solve a question, let him quickly, and with gentlemanly mien endeavoring to be self-possessed, take his stand at the board, read his question distinctly, and with the same reference to rhetorical notation as though he were called out on purpose for the reading of the question. Then let him state his question, giving the reasons for each step as he proceeds; or let him state and solve his question, then return to the commencement, and illustrate the principle, and give the reason for each step in the solution. Then let him pause at the board a moment, for his teacher to propose such questions as he may think proper.

A brief view has now been given of the plan and mode of teaching arithmetic adopted in this system. It is confidently believed, from the long experience the author has had in teaching, that the mode here adopted for presenting the subject of arithmetic, will be found better calculated to induce a fondness for the study; that it unfolds more of the science, and brings out principles more clearly than any other system now before the public. With these views the author submits the work to the candid perusal of all who are interested in the progress of knowledge.

CHARLES G. BURNHAM. DANVILLE, VT., Oct. 18, 1849.

DEFINITIONS, AXIOMS, AND SIGNS.

DEFINITIONS.

A definition is what is meant by a word or phrase. The language of a definition should be so plain as not to be capable of misapprehension.

1. Quantity is any thing which may be multiplied, divided, and measured.

2. Magnitude is that species of quantity which is extended ; i. e. which has one or more of the three dimensions-length, breadth, and thickness. A line is a magnitude, because it has length.

3. Mathematics is the science of quantity.
4. Arithmetic is the science of nunbers.
5. Algebra is a method of computing by letters and other symbols.

6. Geometry treats of lines, surfaces, and solids. Arithmetic, Algebra, and Geometry are those parts of mathematics, on which all the others are founded.

7. A Demonstration is a course of reasoning which establishes a truth.

8. A Proposition is any thing proposed: if to be proved or demonstrated, it is called a Theorem; if to be done, it is called a Problem.

9. A plus quantity is a quantity to be added, and has this sign + before it; thus, +6.

10. A minus quantity is a quantity to be subtracted, and has this sign - before it; thus, – 6.

11. An Equation is a proposition expressing equality between one quantity, or set of quantities, and another, or between different expressions for the same quantity ; thus, 5=3+2.

12. A member of an equation is the quantity or quantities on one side. of the sign of equality.

OBS.-For definitions of terms in more common use in this work, see Art. 54, or Part I.

AXIOMS, An axiom is a self-evident proposition. 1. Things which are equal to the same thing are equal to each other. 2. If equals be added to equals, the wholes will be equal. 3. If equals be taken from equals, the remainders will be equal. 4. If equals be added to unequals, the wholes will be unequal. 5. If equals be taken from unequals, the remainders will be unequal 6. Things which are double of equal things are equal to each other. 7. Things which are halves of the same thing, are equal to each other. 8. The whole is greater than any of its parts. 9. The whole is equal to the sum of all its parts.

SIGNS. Equality is denoted by two horizontal lines. + Addition: as 4+3=7; which signifies that 4 added to 3 equals 7.

x Multiplication: as 4X3=12; which signifies that 4 multiplied by 3 equals 12.

Subtraction: as 4-3=1; which signifies that 3 taken from 4 leaves 1.

)(, ; , 214, Division: as, 2)4(2, and 4-2=2, and 1=2, and ?/4=2. In either case it signifies that 4 divided by 2 equals 2.

::: : Proportion: as, 2 : 4:: 6:12; which is read, 2 is to 4 as 6 is to 12.

Vinculum: as 4+3=7; which is read, the sum of 4 and 3 equals 7, and 4-3=1, is read, the difference of 4 and 3 equals 1.

✓ Radical sign: placed before a number denotes that the square root is to be taken.

42 implies that 4 is to be raised to the second power. 48 implies that 4 is to be raised to the third V implies the third root.

power.

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