AXIOMS.

1. If equal numbers are added to equal numbers, the sums will be equal.

2. If equal numbers are subtracted from equal numbers, the remainders will be equal.

3. If equals be multiplied by equals, the products will be equal.

4. If equals be divided by equals, the quotients will be equal.

5. If two numbers are each equal to the same number, they are equal to each other.

6. If the same number be added to and subtracted from another number, the latter number will not be changed.

17. If a number be both multiplied and divided by the same number, the former number will not be changed.

8. If two numbers be equally increased or diminished, the difference of the resulting numbers will be the same as the difference of the originals.

9. If two numbers are like parts of equal numbers, they are equal to each other.

10. The whole is greater than any of its parts. 11. The whole is equal to the sum of all its parts.

SIGNS.

1. The sign t, called plus, is the sign of addition, and indicates that the number on the right hand is to be added to the one on the left.

2. The sign -, called minus, is the sign of subtraction, and indicates that the number on the right is to be subtracted from that on the left.

3. The sign X, called into, is the sign of multipli. cation, and indicates that the numbers between which it is placed are factors of the same product.

4. The sign +, divided by, the left-hand number to be divided by the right hand.

5. The sign =, equal to, indicates that the numbers between which it is placed are equal.

6. 52, 53, the 2 and 3 placed to the right, a little above a number, indicates the power to which it is to be raised.

ry. V, 0, indicate the extraction of the square and the cube root.

NOTATION AND NUMERATION.

1st. A figure standing alone, as 1, 2, 3, holds the units place, or is of the 1st order, and is read, one, two, three.

2d. A number having two figures, as 14, 26, the righthand figure holds the units place, and the left-hand figure that of tens, and they are read, fourteen, twenty-six.

COR.—The right-hand figure of a number is called units, or the 1st order; the next figure to the left is called tens, or the 2d order; the third figure, hundreds, or the 3d order; the fourth figure, thousands, or the 4th order; and if a number be expressed with the nine figures in order, making 1 the right-hand figure, the figures will express their respective orders; thus,

millions, thousands, units.

co hundreds of
oo tens of

units of

o hundreds of

co hundreds of

one.

21

9 8 g 6

54,32 1 If pointed in periods of three figures each, they may be read as follows: Nine hundred and eighty-seven millions six hundred and fifty-four thousand three hundred and twenty-one,

REM.—The figures designate the orders.
Read the following numbers:

1
".

twenty-one. 321

three hundred and twenty-one. 4,321

four thousand three hundred and twenty-one. 54,321

fifty-four thousand three hundred and twenty-one. 654,321

six hundred and fifty-four thousand three hundred and twenty-one. 7,654,321

seven millions six hundred and fifty-four thousand three hundred and twenty-one. 87,654,321 { eighty-seven millions

six hundred and fifty-four thousand three hundred and twenty-one. 987,654,321 {

nine hundred and eighty-seven millions

six hundred and fifty-four thousand three kundred and twenty-one. REM.—The column of 1's is of the 1st order, the column of 2's is of the 2d order, the 3's the 3d order, the 4's the 4th order, etc.

COR.–The relation of any two consecutive orders is the same, for when in addition the sum of any column reaches 10, the lefthand figure belongs to the next column or order; hence, a table may be formed, thus, 10 units

37 38

1 2 3 4 5 6

40 41

8 9 10 11 12 13 14 15 16 17 18

43 41 45 46 47 48 49 50 51 52 53 54

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This page represents a blackboard with the numbers as high as 72 painted on its margins.

There is also a box containing slips which will cover two, three, four, etc., as high as 12, and numbered accordingly ; one of these the student will take in his hand and apply it to the painted numbers to perform addition or subtraction ; thus, begin at 1 and take a slip marked 2, then 1 and 2 are 3, 3 and 2 ure 5, 5 and 2 are 7, 7 and 2 are 9, etc., counting at least the lefthand column; then, to perform subtraction, begin at the bottom of the 1st column; thus, 36 minus 2 equals 34, 34-2=32, 32—2=30, 30—2=28, etc., until the top is reached ; then taking a slip marked 3, begin with 1 or 2, or first with 1 and then with 2, and return to the top of the column as before, by subtraction ; let this exercise be performed with all the slips, and as the larger numbers are taken, continue the additions to the bottom of the 2d column, and return as before.

For multiplication and division first make a chalk mark after every two figures up to 24, and multiply ; thus, once 2 are 2, twice 2 are 4, 3 times 2 are 6, 4 times 2 are 8, etc.; then the number of divisions is 12 and each division has 2 numbers; .:. 12 is contained twice in 24, or 2 is contained 12 times, 2 is contained once in 2, in 4 twice, in 6 three times, in 8 four times, in 10 five times, in 12 six times, etc. When the student is familiar with multiplication and division by 2, let the numbers be separated into 3's, then 4's, etc., and let each be continued for 12 divisions; when all the divisions have been performed according to the steps, beginning with 2 and ending with 12, a multiplication and division table will be made.

COMPLETE ARITHMETIC.

DEFINITIONS.

1. Arithmetic is the science of numbers.

2. A Unit is a single thing; as, a book, one dollar, or simply one.

3. A Number is a unit or a collection of units; as, one, ten, five books, twenty-five dollars.

4. The numbers used in Arithmetic are all formed by combinations of the ten Arabic characters, called Figures; viz., 0, called zero or naught; 1, called one; 2, two; 3, three; 4, four; 5, five; 6, six; 7, seven; 8, eight; 9, nine.

5. Expressing a number either in writing or figures is called Notation, and reading the expression is called Numeration.

6. When numbers are used without reference to any object, they are called Abstract Numbers; as, five, twenty, etc.; but when they are applied to things, they are called Concrete; as, one book, ten men, four dollars, etc.

sy. When concrete numbers express values of money, weights, measures, time, etc., they are called Denomi