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.

5. According to the census of 1820, Windsor contained 2956 inhabitants, Middlebury 2535, Montpelier 2308, and Burlington 2111; how many inhabitants were there in those four towns?

Operation. 2956 Windsor. 2535 Middlebury. 2308 Montpelier. 2111 Burlington.

9910 Total.

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 $8; 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?

Ana. 12

8192735

214268

21230 7654321 1541320

40212 2121212

25. 2746-590-1001+9976+4321+6633-25067, Ans 26. 39543216-4826382+19181716-63551264, Ans.

5

Four books will evidently cost four times
as much as one book; and to answer the Multiplication.
question by Addition, we should write
down 4 fives, and add them, as at the left
hand. By Multiplication we should pro-
ceed as at the right hand, thus, 4 times 5
are 20. Now these two operations differ

4

Ans. 20 cts.

Ans. 20 cts, only in the form of expression; for we can arrive at the amount of 4 times 5 only by a mental process similar to that 2

at the left hand. Hence, in order to derive any advantage from the use of Multiplication over that of Addition, it is necessary that the several results arising from the multiplication of ion of the numbers below ten, should be perfectly committed to memory. They may be learned from the Multiplication table, page 19. (16)

2. If 1 pound of raisins cost 9 cents, what will 7 pounds cost? 84. 3. There are 24 hours in a day; how many hours are there in 3 days?

Three days will evidently contain

24 hours.

three times as many hours as 1 day, Multiplication.
or 3 times 24 hours; we may there-
fore write down 24 three times, and 23 hours.
add them together, as at the left hand,
or we may write 24 with 3, the num-
ber of times it is to be repeated, under
it, as at the right hand, and say 3 times 4 are 12, (the

LAhs.

8. 72

.

same as 3 fours added together) which are 1 ten and 2 units. We there-
fore write down the 2 units in the place of units, and reserving the 1 ten
to be joined with the tens, we say, 3 times 2 tens are 6 tens, to which we
add the 1 ten reserved, making 7 tens. We therefore write 7 at the left
hand of the 2, in the place of tens, and we have 72 hours, the same as by
Addition. In Multiplication the two numbers which produce the result,
as 24 and 3 in this example, are called factors. The factor which is re-
peated, as the 24, is called the multiplicand; the number which shows
how many times the multiplicand is repeated, as the 3, is called the mul
tiplier; and the result of the operation, as the 72, is called the product.
4. There are 320 rods in a mile; how many rods in 8 miles?
85. 5. A certain orchard consists of 26 rows of trees, and in each row
are 26 trees; how many trees are there in the orchard?

26
26

156 52

Here we find it impracticable to multiply by the whole 28 Operation. at once; but as 26 is made up of 2 tens and 6 units, we may separate them, and multiply first by the units, and then by the tens; thus, 6 times 6 are 36, of which we write down the 6 units, and reserving the 3 tens, we say 6 times 2 are 12, and 3, which was reserved, are 15, which we write down, the 5 in the place of tens, and the 1 in the place of hundreds, and thus find that 6 of the rows contain 156 trees. We now proceed to the 2, and say 2 times 6 are 12; the 2 by which we multiply being 2 tens, it is evident that the 12 are so many tens; but 12 tens are 1 hundred and 2 tens; we therefore write the 2 under the place of tens, which is done by putting it directly under the 2 in the multiplier, and reserve the 1 to be united with the hundreds. We then say 2 times 2 are 4; both these 2's being in the tens' places, their

676

6

10

12

product 4 as nundreds, with which we unite the 1 hundred reserved, making 5 hundreds. The 5 being written at the left hand of the 2 tens, we have 5 hundreds and 2 tens, or 520 for the number of trees in 20 These being added to 156, the number in 6 rows, we have 676 for number of trees in 26 rows, or in the whole orchard.

86. 6. There are in a gentleman's garden 3 rows of trees, and 5 trees in each row; how many trees are there in the whole?

1, 1, 1, 1, 1,

1, 1, 1, 1, 1,

1, 1, 1, 1, 1,

We will represent the 3 rows by 3 lines of 1's, and the 5 trees in each row by 5 1's in each line. Here it is evident that the whole number of 1's are as many times 5 as there are lines, or 3 times 5=15, and as many times

3 as there are columns, or 5 times 3=15. This proves that 5 multiplied by 3 gives the same product as 3 multiplied by 5; and the same may be shown of any other two factors. Hence either of the two factors may be made the multiplicand, or the multiplier, and the product will still be the same. We may therefore prove multiplication by changing the places of the factors, and repeating the operation.

SIMPLE MULTIPLICATION..

87. Simple Multiplication is the method of finding the amount of a given number by repeating it a proposed number of times. There must be two or more numbers given in order to perform the operation. The given numbers, spoken of together, are called factors. factors. Spoke of separately, the number which is repeated, or multiplied, is called the multiplicand; the number by which the multiplicand is repeated, or multiplied, is called the multiplier; and the number produced by the operation is called the product.

RULE.

88. Write the multiplier under the multiplicand, and draw a line below them. If the multiplier consist of a single figure only, begin at the right hand and multiply each figure of the multiplicand by the multiplier, setting down the excesses an carrying the tens as in Addition. (94) If the multiplier con sists of two or more figures, begin at the right hand and multiply all the figures of the multiplicand successively by each figure of the multiplier, remembering to set the first figure of each product directly under the figure by which you are multiplying, and the sum of these several products will be the total product, or answer required. (85)

PROOF.

89. Make the former multiplicand the multiplier, and the former multiplier the multiplicand, and proceed as before; if it be right, the product will be the same as the former. (86)

QUESTIONS FOR PRACTICE.

23. Multiply 848329 by 4009. 24. Multiply 64+7001-103-83 by 18+6. 25. 49×15×17×12×100=how many?