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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 consist ing 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 perids, as in the following

TABLE III.

[Periods. | Sextill. Quintill. Quadrill. Trifl. Billions. Millions. Units. Half pet. th. un. th. un. th. un. th. un. th. un. th. un. cxt. cxu.

Figures. 532,123, 410,861, 232,012, 345,862, 051,234, 525,411,243,673 These two methods agree for the nine first places; but beyond this, the lices take different naines. 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 differenco consisting only in enunciation.

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8. Explain the method of ex-places. pressing number above one hundred. 9. What constitutes the spoken numeration?

10. How is the expression of numbers abridged?

11. What is Notation? How ma.ny methods are there?

24. What is seen by the first Numeration table?

25. What is the rule for reading numbers?

26. How are large numbers, sometimes divided?

27. What is learned from the

12. What are the Roman numer- second, table? .als?

13. Are they in general use? 14. Name the Arabic characters. 15. How are numbers above nine expressed by them?

16. What is the name given to the first place, or right hand figure of a number?

28. What names are repeated in every period?

29. What is the difference between the French and English methods of numeration?

30. What is Numeration?
31. What is Arithmetic?

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 convenien' 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?

36 cents.

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 3 tens of the other, their sum is 5 tens. These two results taken together, are 5 tens and 9 units, or 59,hich 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. Then will the 5 tens and 9 units each stand in its proper place in the answer, making 59.

23 cents.

Ans. 59 cents.

-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.'

Ans. 363 dolls.

Here we write down the numbers as before, and begin with the right hand columm-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.

SIMPLE ADDITION.

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

for 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

82.

SIMPLE ADDITION.

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.

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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.

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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.

14. The difference between two numbers is 1448, and the least number is 2575; what is the greater? Ans. 4023.

15. There are three bags of money, one contains 6462 dollars, one 8224 dollars, and the other 5749 dollars; how many dollars in the three bags? Ans. 20435 dolls.

16. According to the census of the United States in 1820, there were 3995053 free white males, 3866657 free, white females, and 1776289 persons of every other description; what was the whole number of inhabitants at that time? Ans. '9637999.

17. It is 38 miles from Burdington to Montpelier, 47 from Montpelier to Woodstock,

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18. How many days in a common year, there being in January 31 days, in February 28, in March 31, in April 30, in May 31, in June 30, in July 31, in August 31, in September 30, in October 31, in November 30, and in December 31 days? Ans. 365.

19. A person being asked his age, said that he was 9 years old when his youngest brother was born, that his brother was 27 years old when his eldest son was born, and that his son was 16 years old; what was the person's age? Ans. 52 years.

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25. 2746+390+1001+9976+4321+6633=25067, Ans. 26. 99543216+4826852+19181716—63551264, Ans.

2. MULTIPLICATION.

ANALYSIS.

83. We have seen that Addition is an operation by which several numbers are united into one sum. Now it frequently happens that the numbers to be added are all equal, in which case the operation may be abridged by a process called Multiplication.

3. If a book cost 5 cents, what will 4 such books cost?

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