there used in numeration? Thirty-five. Name them! One, two, three, four, five, &c. to twenty; then twenty, thirty, forty, fifty, sixty, &c. to one hundred; then thousand, million, billion, trillion, quadrillion, quintillion, sextillion.

4. In what proportion do numbers increase from unit's place towards the left hand? In ten fold proportion. Why is the Numeration Table made to consist of twelve places of figures rather than any other? Because they make 4 even periods. What is meant by a period? A number expressed by three figures, of which the right hand one signifies so many units, the second so many tens, the third, so many hundreds, &c. Why are three figures called a period? Because, if the number be increased above three places, there is still the same periodical return of value; every third figure to the left will always be hundreds, if ever so far extended.

5. How many methods of numeration are there? Two, the English and French. In what do they differ? The English divide the periods into six figures; the French into three. Which method is practiced? The French. Why? Because it is both convenient and common. Give an example of both methods! What is meant by the Radix of a system? The number by which we reckon. What is the radix of the decimal notation? Ans. TEN.

It is the method of putting together two or more numbers in order to find the whole amount, or total sum of several numbers. This rule may have a few or many given numbers, which are to be brought into one sum of a like denomination. Addition is

of one denomination, when the numbers are all Dollars, Cents, Mills, Acres or Yards.

Ex. Add 11123423138351 and 83543516451442 together.

OPERATION. 11123423138351

EXPLANATION.

To find the answer or amount of the 83543516451442 numbers here given to be added. 94666939589793

RULE.-Begin at figure 2, the unit column, and say, 2 and 1 are 3, which set down directly under 2; proceed to the next column; say, 4 and 5 are 9, (9*); then go to the next, and say, 4 and 3 are .7, (7) ; then go to the next, say, 1 and 8 are 9, (9); then go on to the next column, say, 5 and 3 are 8, which set down directly under the column added; for this must always be done. Finish the opera

tion as commenced.

The operation as now performed is finding the sum of several numbers.

OBS. Always begin to add at the foot of the column on the right hand, and proceed to the left, till all the columns are added.

The figures included in the () must be set down in the operation.

↑ A 0 is not mentioned in adding.

Questions to be answered by the pupils in classes.

What is the first rule in Arithmetic? Addition. How are the fig res placed as to value? Units first; tens next; hundreds next; and so on to millions. How many columns of figures may there be in an addition sum? From two to any number possible.

How are figures in Addition generally placed? One above the other in columns. Does it make any material difference whether the greater or lesser figures be placed first or last? It does not; because 7 and 9 are 16-so are 9 and 7. At which hand do you begin to add? On the right, at the foot of the column. Why? Because units must first be added.

§2. In the foregoing examples (1) no single column amounted to more than nine: but in this §, the principle of carrying for ten, will be introduced and explained.

38976482 To find the sum total of the numbers here 91214509 given to be added.

*130190991

RULE.-Begin at the unit figure 9, and say, 9 and 2are 11; set down 1: one to carry, which added to 8 make 9, set down 9 under 0: none to carry; then say, 5 and 4 are 9, (9): none to carry ; then say, 4 and 6 are 10, (0): 1 to carry, which added to 1 make 2 and 7 are 9, (9): none to carry; now say, 2 and make 11, (1): 1 to carry, which added to 1 make 2, and 8 are 10, (0): 1 to carry, which added to 9 make 10, and 3 are 13, which set down. The operation is now performed.

OBS. The adding of one to the next column, when the figures amount to over 9, (as in the foregoing example) is carrying one for every ten.

Note. The above operation and its explanation virtually illustrates the whole principle of Addition.

Questions to be answered by the pupils in classes.

Low many are 5 and 3? Eight. Why? Because the figure five (5) is for

Set down the whole amount of the last column on the left hand.

fre units or ones, [11111] and three (3) for three units or ones, [111]; which brought into one sum is equal to eight ones, [11111111] written thus. 8.

When a column of figures amount to 9 or a less sum, what figure must be set down? That figure. How many to carry to the next column? None. Why? For the column does not amount to any tens. It takes ten ONES UT units to make one 10. Suppose a column, when added amounts to 12; what must be set down? 2. Why? Because there are two units and one ten in 12. How many must be carried or added to the first figure of the next column? 1. Why? Because the one stands in tens' place, and must be located in that denomination; Ten in units' place make one in tens' place; and ten tens make one in the place of hundreds.

How many units and tens in 16 integers? Six units and one ten. Why? Because 6 occupies the first place in numeration. And because the 1 stands in tens' place, or the second place of numerical notation.

Why do you carry one for every TEN? Because numbers increase in a tenfold proportion; and because 10, in an inferior column, is just equal to one in a superior column.

3. Let the pupil proceed to the performance of the following operations. The answers are given in the Key, which ought not to be consulted by the pupil till an answer shall have been obtained.

The pupil may write down in figures the following sums, and find their amount.

1. Add Four thousand three hundred fifty-four-Two hundred sixty-three-Nine hundred twenty-one, and sixty-eight, together. Ans. Five thousand six hundred and six.

2. Find the amount of Nine hundred twenty-six-Six thousand four hundred sixty-three; and One hundred sixty-nine. Ans. Seven thousand five hundred fifty-eight.

3. How do you set down and add eleven thousand, eleven hundred, eleven? Ans. 12111 4. How do you write fourteen thousand, fourteen hundred, fourteen, and add it?

Ans. 15414

DEFINITION OF TERMS IN ADDITION.

A definition is an explanation of what is meant by any word or phrase.

It is essential to a complete definition, that it perfectly dis tinguishes the thing defined from every thing else.

Addition, the bringing of numbers or quantities together to find their amount.

Amount, the sum total of two or more numbers, brought inte

one.

Answer, the sum total of numbers, the result of an operation. * Page 230,

Carry, the adding of one denomination to another according to its ratio.

Column, a perpendicular line of figures. A line is the dis tance between two points, and has length only.

Row, a number of figures ranged in a right line.
Figure, the representation of one or more units.
Unit, the least
whole number.

Number, one or more units

Adding, the act of putting numbers together.

The Rule for performing operations in Addition may be given thus:

1. Begin at the right hand column, and add together, upward, all the figures in it, and place the amount, if less than ten, under that column.

2. If the amount be just ten, place a cipher there, and carry one to the next left hand column.

3. If more than ten, or two or more even tens, set down all there is over, and carry one for each even ten to the next left hand column.

4. Proceed in this way through all the columns, and set down the full amount under the left hand column.

§ 4. The design of the following examples is to discover whether the pupil has been thorough in the solutions of previous questions, and to prevent him from transcribing the answers. Find the sum of these two numbers, viz:

Amt. of Nos. 5, and 6. 3893654. Amt. of Nos. 7, and 8. 4933108.