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place, which is the place of tens, it must be followed by one cipher, thus, 60, (sixty.) The lowest order, or units, are five, represented by a single character, thus, 5, (five.)

We may now combine all these parts together, first writing down the five units for the right hand figure, thus, 5; then the six tens (60) on the left hand of the units, thus, 65; then the three hundreds (300) on the left hand of the six tens, thus, 365, which number, so written, may be read three hundred, six tens, and five units; or, as is more usual, three hundred and sixty-five.

13. Hence it appears that figures have a different value according to the PLACE they occupy, counting from the right hand towards the left.

Hund.
Tens.
Units,

Take for example the number 3 3, made by the same figure three times repeated. The 3 on the right hand, or in the first place, signifies 3 units; the same figure, in the second place, signifies 3 tens, or thirty; its value is now increased ten times. Again, the same figure, in the third place, signifies neither 3 units, nor 3 tens, but 3 hundreds, which is ten times the value of the same figure in the place immediately preceding, that is, in the place of tens; and this is a fundamental law in notation, that a removal of one place towards the left increases the value of a figure TEN TIMES.

Ten hundred make a thousand, or a unit of the fourth order. Then follow tens and hundreds of thousands, in the same manner as tens and hundreds of units. To thousands succeed millions, billions, &c., to each of which, as to units and to thousands, are appointed three places,* as exhibited in the following examples :

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• This is according to the French method of counting. The English, after hundreds of millions, instead of proceeding to billions, reckon thousands, tens and hundreds of thousands of millions, appropriating six places, instead of three, to millions, billions, &c.

EXAMPLE 2d.

3, 174, 5 92, 837, 4 6 3,

5 1 2

To facilitate the reading of large numbers, it is frequently practised to point them off into periods of three figures each, as in the 2d example. The names and the order of the periods being known, this division enables us to read numbers, consisting of many figures as easily as we can read three figures only. Thus, the above examples are read 3 (three) Quadrillions, 174 one hundred seventy-four) Trillions, 592 (live hundred ninety-two) Billions, 837 (eight hundred thirty-seven) Millions, 463 (four hundred sixty-three) Thousands, 512 (tive hundred and twelve.)

After the same manner are read the numbers contained in the fol. lowing

NUMERATION TABLE.

Those words at the head of the table are applicable to any sum or number, and must be committed perfectly to memory, so as to be readily applied on any occasion.

Hundreds of Millions.
Tens of Millions.
Hundreds of Thousands.
Tens of Thousands.
Millions.
Thousands.
Hundreds.

Tens.
v Units.

• 8 6

.4 3 2 ... 7 0 5 4 ...86 200

• 9 0 0 3 7 1 • • 50 86000 •1 0 3 0 2 0 7 0 80 6 1 0 5 4 0 9

Of these characters, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, the nine first are sometimes called significant figures, or digits, in distinction from the last, which, of itself, is of no value, yet, placed at the right hand of another figure it increases the value of that figure in the same tenfold proportion as if it had been followed by any one of the significant figures.

Note. Should the pupil find any difficulty in reading the following numbers, let him first transcribe them, and point them off into periods. 5768 52831209

286297314013 34120 175264013

5203845761204 701602 3456720834

13478120673019 539285 25037026531

341246801734526 The expressing of numbers, (as now shown,) by figures, is called

Notation. The reading of any number set down in figures, is called Numeration.

After being able to read correctly all the numbers in the foregoing table, the pupil may proceed to express the following numbers by figures :1. Seventy-six. 2. Eight hundred and seven. 3. Twelve hundred, (that is, one thousand and two hundred.) 4. Eighteen hundred. 5. Twenty-seven hundred and nineteen. 6. Forty-nine hundred and sixty. 7. Ninety-two thousand and forty-five. 8. One hundred thousand. 9. Two millions, eighty thousands, and seven hundreds..

10. One hundred millions, one hundred thousand, one hundred and one.

11. Fifty-two millions, six thousand, and twenty. 12. Six billions, seven millions, eight thousand, and nine hundred.

13. Ninety-four billions, eighteen thousand, one hundred and seventeen.

14. One hundred thirty-two billions, two hundred millions, and nine.

15. Five trillions, sixty billions, twelve millions, and ten thousand.

16. Seven hundred trillions, eighty-six billions, and seven milLions.

ADDITION
OF SIMPLE NUMBERS.

14. 1. James had 5 peaches, his mother gave him 3 peaches more; how many peaches had he then ?

2. John bought a slate for 25 cents, and a book for 8 cents; how many

cents did he give for both ? 3. Peter bought a wagon for 36 cents, and sold it so as to gain 9 cents; how many cents did he get for it?

4. Frank gave 15 walnuts to one boy, 8 to another, and had 7 left; how many walnuts had he at first?

5. A man bought a chaise for 54 dollars; he expended 8 dollars in repairs, and then sold it so as to gain 5 dollars; how many dollars did he get for the chaise ?

6. A man bought 3 cows; for the first he gave 9 dollars, for the second he gave 12 dollars, and for the other he gave 10 dollars; how many dollars did he give for all the cows ?

7. Samuel bought an orange for 8 cents, a book for 17 cents, a knife for 20 cents, and some walnuts for 4 cents; how many cents did he spend ?

8. A man had 3 calves worth 2 dollars each, 4 calves worth 3 dol. lars each, and 7 calves worth 5 dollars each; ho'y many calves had he?

9. A man sold a cow for 16 dollars, some corn fo. 20 dollars, wheat for 25 collars, and butter for 5 dollars; how many dollars must he receive ?

The putting together two or more numbers, (as in the foregoing examples,) so as to make one whole number, is called Addition, and the whole number is called the sum or amount.

10. One man owes me 5 dollars, another owes me 6 dollars, another 8 dollars, another 14 dollars, and another 3 dollars; what is the amount due to me ?

11. What is the amount of 4, 3, 7, 2, 8, and 9 dollars ?

12. In a certain school, 9 study grammar, 15 study arithmetic, 20 attend to writing, and 12 study geography; what is the whole number of scholars ?

SIGNs. A cross, t, one line horizontal and the other perpendicular, is the sign of addition. It shows that numbers with this sign between them are to be added together. It is sometimes read plus, which is a Latin word signifying more.

Two parallel, horizontal lines, =, are the sign of equality. It signifies that the number before it is equal to the number after it. Thus, 5+3=8 is read 5 and 3 are 8; or, 5 plus (that is, inore) 3 is equal to 8. In this manner let the pupil be instructed to commit the following

ADDITION TABLE.
2 0= 2 4 +0= 4 6 +0= 6 8

0= 8
2 I= 3 4
5 6+1= 7 8

1= 9
2 2 = 4

2 = 6 6 +2= 8 8 - 2 = 10
2
5 4
6 3 9 8

3= 11
2
6
8 6-7 4 = 10

= 12
5 = 7 4

9 6 5 11

8 +5 = 13 2 + 6 = 8 4 +6= 10 6-6= 12 8 +6= 14 2 +7 9 4 +7=11 6 7 13 8

7 15 28=10 4 +8= 12 6 8 = 14 8

8 = 16 2 9 11 4 9 = 13 6

8

9 3 +0= 3 5 +0= 5 7 +0= 7 9

9
3 tl= 4 5 += 6 7 1= 8 9 1 10
3+2= 5 5+2= 7 7

9 9 +2= 11
3+3= 6 5 +3= 8 7 +3 10 9 +3=12
3+4=
7 5 +4= 9 7

11 9 4 = 13
3.5= 8 5 + 5 10

12 9 +5 = 14 3 +6= 9 5 6 = 11

6 13 9 6 3 +7=10 5+7= 12 777 14 9

7 = 16
3 +8=11 5+8=13

8 15 9 8= 17
3
9 = 12 5 9 = 14 7 9 16 9

9 = 18

15

S17

7+

15

5 +9= how many ?
8+7= how many ?
4 +3 +2= how many ?
6 +4 -45 = how many ?

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T 5. When the numbers to be added are small, the addition is readily performed in the mind; but it will frequenily be more convenient, and even necessary, to write the numbers down before adding them.

13. Harry had 43 cents, his father gave him 25 cents more; how many cents had he then ?

One of these nunhers contains 4 tens and 3 units. The other number contains 2 tcns and 6 units. To unite these two numbers together into one, write them down one under the other, placing the units of one number directly under units of the other, and the tens of one number directly under tens of the other, thus :

43 cents. Having written the numbers in this manner, draw a 23 cents. line underneath.

We then begin at the right hand, and add the 5 43 cents.

units of the lower number to the 3 units of the upper 25 cents.

number, making 8 units, which we set down in unit's

place. 8

We then proceed to the next column, and add the 43 cents. 2 tens of the lower number to the 4 lens of the upper 25 cents. number, making 6 tens, or 60, which we set down in

ten's place, and the work is done. Ans.68 cents.

It now appears that Harry's whole number of cents is 6 tens and 8 units, or 68 cents; that is, 43 + 25 = 68.

14. A farmer bought a chaise for 210 dollars, a horse for 70 dollars, and a saddle for 9 dollars; what was the whole amount ?

Write the numbers as before directed, with units under units, tens under tens, &c.

OPERATION. Chaise, 210 dollars, Add as before. The units will be 9, the Horse, 70 dollars. tens 8, and the hundreds 2; that is, 210 to Saddle, 9 dollars. 70 +9=289. Ansioer, 239 dollars.

After the same manner are performed the following examples: 15. A man had 15 sheep in one pasture, 20 in another pasture, and 13 in another ; how many sheep hod be in the three pastures ? 16 to 20 + 143 = how many ?

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