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by Columbus, 1492; Vasques de Gama's discovery of the route to the East Indies by the Cape of Good Hope, 1497; commencement of the Reformation, 1517; spinning-wheel invented, 1530; Copernicus died, 1543; telescopes invented, 1590; University of Dublin founded, 1591; English East India Company established, 1600; Decimal Fractions invented, 1602; thermometers invented, and satellites of Jupiter discovered, 1610; Logarithms published by Napier, 1614; circulation of the blood discovered by Harvey, 1619; barometers invented, 1643; air-pump invented, 1654; Newtonian philosophy published, 1686; Cornwallis surrendered to the American army in 1781; the Constitution of the United States submitted to Congress, 1787; union of Great Britain and Ireland, 1801; battle of Waterloo, 1815.

What is subtraction?

Questions.

Repeat the rule for performing the operation.
What are the methods of proof.

Examples in Addition and Subtraction.

Ex. 1. A man has five apple-trees, of which the first bears 157 apples, the second 264, the third 305, the fourth 97, and the fifth 123. He sells 428 apples; 186 are stolen. How many has he left for his own use? Ans. 332.

Ex. 2. Out of an army of 57068 men, 9503 are killed in -battle; 586 desert to the enemy; 4794 are taken prisoners; 1234 die of their wounds on the passage home; 850 are drowned. How many return alive to their own country? Ans. 40101.

Ex. 3. A man, travelling from New-York to Washington, went the first day 35 miles, the second day 60 miles, the third day 59 miles, and going the fourth day 36 miles, he was within 38 miles of Washington. What is the distance between New-York and Washington; and how far from the latter city was the traveller at the end of the third day?

Ans. From N. Y. to W. 228 miles, and 74 miles. Ex. 4. A man, at the beginning of the year, finds himself worth 123078 dollars. In the course of the year, he gains by trade $8706; but spends in January $237, in February $301, and in each succeeding month (being 10 months) as

much as in the first two. What was the state of his affairs Ans. $125866.

at the end of the year?

Ex. 5. In the South District of the state of New-York, consisting of six counties, the population in 1820 was 219457; the population in the Middle District, consisting of nine Counties, 237650; in the East District, consisting of eleven counties, 243826; and in the West District, consisting of twenty-four counties, 1372812. Required the excess of the population of the West District over the population of the other three Districts. Ans. 671879.

Ex. 6. A merchant bought 756 barrels of flour for $3560, 259 barrels for $1250; and he sold afterwards 1000 barrels for $5000. How much does he gain, and how many barrels has he left? Ans. He gains $190, and 15 barrels unsold.

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Ex. 7. The public debt of the United States in 1791, was $75463467; from 1791 till 1812, the debt was reduced $38806535; from this date till 1816, it was increased $86359443; and from this date till 1820, it was reduced $31336285. What was the public debt in the year 1820 ? Ans. $91680090. Ex. 8. James Cook, the navigator, was born in 1728, and he was killed in 1779. How old was he at his death? Ans. 51 years.

Ex. 9. Alexander Pope, the poet, was 55 years old when he died in 1744. In what year was he born?

Ans. 1689, Ex. 10. William Penn, the founder of Pennsylvania, landed first at New-Castle, in the year 1682: How many years since his landing till the present time (1827?) Ans. 145 years.*

MULTIPLICATION OF WHOLE NUMBERS.

20. Multiplication of whole numbers is but an abridged method of addition, employed where we have occasion to add the same number repeatedly to itself.

Of the two numbers multiplied together, and called

* Chronology wil! furnish the teacher with an indefinite variety of examples; but it is to be observed, in general, that pains should be taken to give the child a clear conception of the terms employed in a question, before he is called to solve it: and that the first illustrations of the use of arithmetical rules should be borrowed from the objects with which he is most familiar, and proposed in low numbers. The great advantage of an early application to arithmetic is the exercise which it affords to the thinking faculty.

by the common name of factors, the multiplicand is that number which we want to add repeatedly to itself; and the multiplier expresses the number of times that the former is to be repeated in that addition: the sum required is called the product.

Thus, by the product of 7 multiplied by 4 we are really to understand the sum of 4 sevens, or 7+7+7+7=28.

21. The product of any two numbers is the same, whichever of them be made the multiplier.

For instance, if we multiply 8 by 5, we shall have the same product as if we multiply 5 by 8. It is by no means self-evident, that the sum of 5 eights must be the same with the sum of 8 fives, or that 8+8+8+8+8=5+5+5+5 +5+5+5+5; which is the meaning of the proposition. However, it admits of a very easy proof from the following illustration. Suppose 5 rows of 8 counters, regularly disposed under each other. Whatever way we count them, the total amount of the number must be the same. counting them one way, we have 5 times eight; and counting them another way, it is plain that we have 8 times five counters. It is obvious that a similar proof would be applicable to any higher numbers.

But

The sign of multiplication is X, interposed between the factors; and is to be carefully distinguished from the sign of addition. Thus, 12 x 8, or 8x12, denotes the product of 8 and 12.

22. The product of any two numbers is equal to the sum of all the products obtained by multiplying all the parts, into which either is divided, by the other, or by each of the parts into which the other is divided.

Thus, if we suppose 8 to be divided into the parts 4, 3, and 1; the product of 5 times & will be equal to the sum of the three products, 5 times 4, 5 times 3, and 5 times 1. And if we suppose the multiplier 5 also divided into the two parts 3 and 2, the product of 5 times 8 will be equal to the six products obtained by multiplying each of the three component parts of the multiplicand by each of the two component parts of the multiplier. The truth of this will appear very plain, by employing the same illustration that was adduced in § 21. In the 5 rows of 8 counters, aptly representing 5 times 8, let us suppose, first, two lines drawn D

downwards, dividing each row of eight counters into the three parts 4, 3, and 1. It is then plain that the whole set of 5 times 8 counters is divided into three sets of 5 times 4, 5 times 3, and 5 times 1. Then supposing a line drawn across and dividing each row of 5 counters into 3 and 2, it is plain that each of the three former sets will be divided into two, 3 times 4 and twice 4, 3 times 3 and twice 3, 3 times 1 and twice 1: so that the sum of these six sets is equal to the one set of 5 times 8 counters. This proof is exhibited to the eye in the subjoined scheme.

0000 000 0

0000 000 0

0000 0000

0000 0000

0000 0000

And it is plainly applicable to any other numbers, divided into any parts whatsoever. Thus, if we suppose 17 broken into the four parts 6, 5, 4, and 2; and 9 broken into the three parts 4, 3, and 2; the product of 9 times 17 must be equal to the sum of each of the twelve products obtained by multiplying each of the four parts of the multiplicand by each of the three parts of the multiplier: that is, 17×9=' (24+20+16+8)+(18+15+12+6)+(12+10+8+ 4), or 153=68+51 +34. With the principle brought forward in this section the student cannot be too familiar; as it is the foundation of common multiplication and algebraic, as well as fruitful in the most important inferences.

23. If the multiplier be the product of any two known numbers, we may employ a successive multiplication by the factors of which the multiplier is the product.

Thus, if we want to multiply any number by 54, we may multiply it by 9, and that product by 6; for, 6 times 9 being 54, when we first find a number that is 9 times the multiplicand, and then multiply that number by 6, the product must be 6 times 9 times, or 54 times the multiplicand.

24. The product of any number multiplied by 10, 100, 1000, &c. is obtained at once by annexing one, two, three, &c. ciphers to the multiplicand on the right hand.

Thus, the product of 327 multiplied by 1000 is 327000 ;

for each digit of the multiplicand is increased in value 1000 times. And combining the principle of the last section, it is plain that if the multiplier be 20, 300, 4000, &c. we may obtain the product by annexing one, two, three, &c. ciphers, and then multiplying by 2, 3, 4, &c. Thus, 4296 × 700= 429600 X 7.

From the principle stated in § 23, it is manifest that we can find the product of any two numbers: for however great the factors, they may be broken into parts not exceeding The product of all such parts are furnished by the multiplication table, which should be committed to memory.

12.

MULTIPLICATION TABLE.*

twice 3 times 4 times 5 times 6 times 7 times
1=2 1=3 1=4 1=5 1=6 1=7
2=4 2=6 2=8 2=10 2=12 2=14
3=6 3=9 3=12 3=15 3=18 3=21
4=8 412 416 4 20 4=24| 4=28
5=10 515 5=20 525 530 5=35
6=12 6=18 6=24 6=30 6=36 6=42
7=14 7=21 7=28 7=35 7=42 7=49
8-16 8-24| 8=32 8=40 848 8=56
9-18 9-27 9-36 945 954 9=63
10=2010=30|10=40|10=5010=60|10=70
11=2211=3311=44|11=5511=6611=77
12=2412=3612=48|12=60|12=7212=84|

8 times 9 times 10 times 11 times 12 times
1=8 1=9 1=10 1=11 1=12

[blocks in formation]

9=72 9-81 9=90 9=99 9=108 10 80 10 90 |10=1001011010=120 11=8811=99 11=11011=121|11=132 12-9612-108 12=120 1213212=144

*Though the part of the multiplication table given in the text is quite eough for the pupil to commit to memory at first; yet, at rhe has

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