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Ipswich 3000, Lynn 9369, Lynnfield 707, Manchester 1355, Marblehead 5575, Methuen 2251, Middleton 657, Newbury 3789, Newburyport 7161, Rockport 2650, Rowley 1203, Salem 15082, Salisbury 2739, Saugus 1098, Topsfield 1059, Wenham 689, West Newbury 1560? Ans. 94,989.

52. How many were the members of Congress in 1846, there being 2 Senators from each State, and Maine sending 7 Representatives, New Hampshire 4, Massachusetts 10, Rhode Island 2, Connecticut 4, Vermont 4, New York 34, New Jersey 5, Pennsylvania 24, Delaware 1, Maryland 6, Virginia 15, North Carolina 9, South Carolina 7, Georgia 8, Alabama 7, Mississippi 4, Louisiana 4, Tennessee 11, Kentucky 10, Ohio 21, Indiana 10, Illinois 7, Missouri 5, Arkansas 1, Michigan 3, Florida 1, Texas 2 ?

Ans. 282.

53. According to the census of 1840, Maine had 501,793 in habitants, New Hampshire 284,574, Massachusetts 737,699, Rhode Island 108,830, Connecticut 309,978, Vermont 291,948 New York 2,428,921, New Jersey 373,306, Pennsylvania 1,724,033, Delaware 78,085, Maryland 469,232, District of Columbia 43,712, Virginia 1,239,797, North Carolina 753,419, South Carolina 594,398, Georgia 691,392, Kentucky 779,828, Tennessee 829,210, Ohio 1,519,467, Indiana 685,866, Mississippi 375,651, Missouri 383,702, Illinois 476,183, Louisiana 352,411, Alabama 590,756, Michigan 212,267, Arkansas 97,574, Florida 54,477, Wisconsin 30,945, Iowa 43,112, and on board U. S. vessels 6,100. What was the whole number of inhabitants? Ans. 17,068,666.

SECTION III.

SUBTRACTION.

SUBTRACTION teaches to find the difference between two numbers by taking the less from the greater.

From 935
Take 673

262

OPERATION. In this question, we take 3 units from 5 units and 2 units remain, which we write down under units, as the first figure in the answer. In attempting to take the 7 tens from 3 tens we find a difficulty, as 7 cannot be taken from 3. therefore borrow 1 (hundred) from the 9 (hundred), which being equal to 10 tens, we add it to the 3 tens in the upper line, making 13 tens; from which we take 7 tens, and 6 tens re

We

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main, which we write down under the place of tens. then proceed to the hundreds. As we have borrowed 1 from the 9 hundreds, the 9 is too large by 1. We must therefore take the 6 (hundreds) from 8 hundreds and there will remain 2 (hundreds). We therefore write down the 2 in the place of hundreds. Or, because the 9 is too large by 1, we may add 1 to the 6, and say 7 from 9 and 2 will remain. Hence the following

RULE.

Place the less number under the greater; units under units, tens under tens, &c. Begin with the units, and if the lower figure be smaller than the one above it, write the difference below. But, if the upper figure be less than the lower, then add ten to the upper one, and write the difference between the sum thus obtained and the lower figure. Then carry or add one to the lower figure of the next column, and proceed as before, till all the numbers are subtracted, and the result will be the difference.

NOTE. The upper number is called the Minuend, from the Latin word minuendum, signifying to be made less; and the lower one the Subtrahend, from subtrahendum, to be taken away. The result is the Remainder.

PROOF.

Add the remainder to the subtrahend, and, if their sum be like the minuend, the work may be considered correct.

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44. Sir Isaac Newton was born in the year 1642, and he died in 1727; how old was he at the time of his decease?

Ans. 85 years.

45. Gunpowder was invented in the year 1330; how long was this before the invention of printing, which was in 1441 ? Ans. 111 years.

46. The mariner's compass was invented in Europe in the

year 1302; how long was this before the discovery of America

by Columbus, which happened in 1492? 47. What number is that, to which if sum will be one million?

Ans. 190 years.

6956 be added, the Ans. 993044.

48. A man bought an estate for seventeen thousand five hundred and sixty-five dollars, and sold it for twenty-nine thousand three hundred and seventy-five dollars. Did he gain or lose, and how much? Ans. Gained $11810.

49. Bought a pair of oxen for 85 dollars, a horse for 126 dollars, three cows at 25 dollars apiece; and sold the whole for three hundred dollars; how much did I gain? Ans. $14. 50. Bonaparte was declared emperor in 1804; how many years since?

51. The union of the government of England and Scotland was in the year 1603; how long was it from this period to the time of the declaration of the independence of the United States? Ans. 173 years.

52. Jerusalem was taken and destroyed by Titus in the year 70; how long was it from this period to the time of the first Crusade, which was in the year 1096 ? Ans. 1026 years.

SECTION IV.

MULTIPLICATION.

MULTIPLICATION is the repetition of a number any proposed number of times. It consists of three parts, the Multiplicand, or number to be multiplied; the Multiplier, or number by which to multiply; and the result, which is called the Product. The Multiplicand and Multiplier are called factors.

RULE.

Place the larger number uppermost for the multiplicand, and the smaller number under it for a multiplier, arranging units under units, tens under tens, &c. Then multiply each figure of the multiplicand by each figure of the multiplier, beginning with the right-hand figure, and carrying for every ten as in addition. If the multiplier consists of more than one figure, the right-hand figure of each product must be placed directly under the figure of the multiplier that produces it, which will cause the successive products to recede each one place to the left. The sum of the several products will be the whole product required.

NOTE 1. - When there are ciphers between the significant figures of the multiplier, pass over them in the operation, and multiply by the

significant figures only, remembering to set the first figure of the product directly under the figure of the multiplier that produces it. See Ex. 15. NOTE. 2.If there are ciphers at the right hand either of the multiplier or multiplicand, or of both, they may be neglected to the close of the operation, when they must be annexed to the product.

PROOF.

The correctness of the result in Multiplication may be conveniently ascertained in three ways; viz., by Division, by Multiplication, or by casting out the nines.

According to the first method,* divide the product by the multiplier; and, if the work is right, the quotient will be equal to the multiplicand.

According to the second method, take the multiplier for the multiplicand and the multiplicand for the multiplier, and proceed according to the rule for multiplication; and, if the work be right, the product will be the same as by the former operation.

According to the third method, begin at the left hand of the multiplicand, and add together its successive figures towards the right, till the sum obtained equals or exceeds the number 9. If it equals it, drop the nine, and begin to add again at this point, and proceed till you obtain a sum equal to or greater than nine. If it exceeds nine, drop the nine as before, and carry the excess to the next figure, and then continue the addition as before. Proceed in this way till you have added all the figures in the multiplicand and rejected all the nines contained in it, and write the final excess at the right hand of the multiplicand. Proceed in the same manner with the multiplier, and write the final excess under that of the multiplicand. Multiply these excesses together and place the excess of nines in their product under the other excesses. Then proceed to find the excess of nines in the product obtained by the original operation, and, if the work be right, the excess thus found will be equal to the excess contained in the product of the above excesses of the multiplicand and multiplier. See Example 15.

NOTE. This method of proof, though perhaps sufficiently sure for common purposes, is not always a test of the correctness of an operation. Cases will sometimes occur in which the excesses above named will be equal, when the work is not right.

* As the pupil is presumed not to be acquainted with Division, he will pass over this method of proof for the present. It is placed here as a method important to be known, and because there seems to be no better place for it, though it presupposes an acquaintance with a rule yet to be learned.

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