higher order, is called the ten's place. Ten tens, or one hundred, is written, 100, two hundred, 200, and so on to nine hundred, 900, and the intermediate numbers are expressed by writing the excesses of tens and units in the tens' and units' places, instead of the ciphers. Two hundred and twenty-two is written, 222. Here we have the figure 2 repeated three times, and each time with a different value. The 2 in the second place denotes a number ten times greater than the 2 in the first;, and the 2 in the third, or hundreds' place, denotes a number ten times greater than the 2 in the second, or ten's place; and this is a fundamental law of Notation, that each removal of a figure one place to the left hand increases its value ten times.

74. We have seen that all numbers may be expressed by repeating and varying the position of ten figures. In doing this, we have to consider these figures as having local values, which depend upon their removal from the place of units. These local values are called the names of the places: which may be learned from the following TABLE 1.

coSextillions.

Hund. of Quint.
Tens of Quint.
Quintillions.
Hund. of Quad.
Tens of Quad.
Quadrillions.
co Hund. of Trill.
Tens of Trill.
coTrillions.
Hund. of Bill.
Tens of Bill.
Billions.

Hund. of Mill.
Tens of Mill.
Millions.
cHund. of Thou

Tens of Thou.
Thousands.
Hundreds.
coTens.

Units.

6 7

By this table it will be seen that 2 in the first place denotes

simply 2 units, that 3 in the second place denotes as many tens as there are simple units in the figure, or 3 tens; that 2 in the third place denotes as many hundreds as there are units in the figure, or 2 hundreds; and so on.

Hence to read any number, we have only to observe the following RULE.-To the simple value of each figure join the name of its place, beginning at the left hand, and reading the figures in their order towards the right.

The figures in the above table would read, three sextillions, four hundred fifty-six quintillions, seven hundred fifty-four quadrillions, three hundred seventy-eight trillions, four hundred sixty-four billions, nine hundred seventy-four millions, three hundred one thousand, two hundred thirty-two.

75. In reading very large numbers it is often convenient to divide them into periods of three figures each, as in the following

532, 123,

410,864,232,012, 345, 862,051,234, 525, 411, 243, 673.

By this table it will be seen that any number, however large, after dividing it into periods, and knowing the names of the periods, can be read with the same ease as one consisting of three figures only; for the same names, (hundreds, tens, units,) are repeated in every period, and we have only to join to these, successively, the names of the periods. The first, or right hand period, is read, six hundred seventy-three -units, the second, two hundred forty-three thousands, the the third, four hundred eleven millions, and so on.

76. The foregoing is according to the French numeration, which, on account of its simplicity, is now generally adopted in English books. In the older Arithmetics, and in the two first editions of this work, a period is made to consist of six figures, and these were subdivided into half periods, as in the following

TABLE III.

Periods. Sextill. Quintill. Quadrill. Trill. Billions. Millions. Units. th. un. th. un. th. un. ext. cxu.

Half per. th. un. th. un. th. an. Figures. 532,123, 410,864, 232,012,345,862, 051,234, 525,411,243,673 These two methods agree for the nine first places; but beyond this, the places take different names. Five billions, for example, in the former method, is read five thousand millions in the latter. The principles of notation are, notwithstanding, the same in both throughout-the difference consisting only in enunciation.

REVIEW.

SECTION II.

SIMPLE NUMBERS.

77. Numbers are called simple, when their units are all of the same kind, as men, or dollars, &c.

1. ADDITION.
ANALYSIS.

78. 1. How many cents are 3 cents and 4 cents?

Here are two collections of cents, and it is proposed to find how large a collection both these will make, if put together. The child may not be able to answer the question at once; but having learned how to form numbers by the successive addition of unity, (2, 72,) he will perceive that he can get the answer correctly, either by adding a unit to 4 three times, or a unit to 3 four times, (7). In this way he must proceed, till, by practice, the results arising from the addition of small numbers are committed to memory; and then he will be able to answer the ques

tions which involve such additions almost instantaneously. But when the numbers are large, or numerous, it will be found most convenient to write them down before performing the addition.

2. A boy gave 36 cents for a book, and 23 cents for a slate, how many cents did he give for both?

Here the first number is made up of 3 tens and 6 units, and the second of 2 tens and 3 units. Now if we add the 3 units of one with the 6 units of the other, their sum is 9 units, and the 2 tens of one added to the 8tens of the other, their sum is 5 tens. These two results taken together, are 5 tens and 9 units, or 59, which is the number of cents given for the book and slate. The common way of performing the above operation is to write the numbers under one another, so that units shall stand under units, and tens under tens, as at the left hand. Then begin at the bottom of the right hand column, and add together the figures in that column; thus, 3 and 6 are 9, and write the 9 directly under the column. Proceeding to the column of tens, we say, 2 and 3 are 5, and write the 5 directly under the column of tens. will the 5 tens and 9 units each stand in its proper place in the answer, making 59.

36 cents.
23 cents.

Ans. 59 cents.

Then

3. If a man travel 25 miles the first day, 30 the next, and 33 the next, how far will he travel in the three days? Ans. 88 miles. 79. 4. A man bought a pair of horses for 216 dollars, a sleigh for 84 dollars, and a harness for 63 dollars; what did they all cost him?

216 dolls.
84 dolls.
63 dolls.

Here we write down the numbers as before, and begin with the right hand column-3 and 4 are 7, and 6 are 13; but 13 are 1 ten and 3 units; we therefore write the 3 under the column of units, and carry the 1 ten to the column of tens, saying, 1 to 6 are 7, and 8 are 15, and 1 are 16. But 16 tens are 1 hundred and 6 tens; we therefore write the 6 under the column of tens, and carry the 1 into the column of hundreds, saying, 1 to 2 are 3, which we write down in the place of hundreds, and the work is done. From what precedes, the scholar will be able to understand the following definition and rule.

Ans. 363 dolls.

SIMPLE ADDITION.

80. Simple Addition is the uniting together of several simple numbers into one whole or total number, called the sum,

or amount.

RULE.

81. Write the numbers to be added under one another, with units under units, tens under tens, and so on, and draw a line below them. Begin at the bottom, and add up the figures in the right hand column:-if the sum be less than ten, write it below the line at the foot of the column; if it be ten, or an exact number of tens, write a cipher, and carry the tens to the next column; or if it be more than ten, and not an exact number of tens, write down the excess of tens, and carry the tens as above. Proceed in the same way with

the columns of tens, hundreds, &c. always remembering, that ten units of any one order, are just equal to one unit of the next higher order.

PROOF.

82. Begin at the top, and reckon each column downwards, and if their amounts agree with the former, the operation is supposed to have been rightly performed.

NOTE. No method of proving an arithmetical operation will demonstrate the work to be correct; but as we should not be likely to commit errors in both operations, which should exactly balance each other, the proof renders the correctness of the operation highly probable.

QUESTIONS FOR PRACTICE.

9. In a certain town there are 8 schools, the number of scholars in the first is 24, in the second 32, in the third 28, in the fourth 36, in the fifth 26, in the sixth 27, in the seventh 40, and in the eighth 58; how many scholars in all the schools? Ans. 251.

10. Sir Isaac Newton was born in the year 1642, and was 85 years old when he died; in what year did he die? Ans. 1727.

11. I have 100 bushels of wheat, worth 125 dollars, 150 bushels of rye, worth 90 dollars, and 90 bushels of corn, worth 45 dollars; how many bushels have I, and what is it worth?

Ans. 340 bush. worth 260 dolls.

12. A man killed 4 hogs, one weighed 371 pounds, one 510 pounds, one 472 pounds, and the other 396 pounds; what did they all weigh?

Ans. 1749 pounds.

13. The difference between two numbers is 5, and the least number is 7; what is the greater? Ans. 12.