3. Water-mills were invented in the year 555 after Christ; windmills 744 years after water-mills; pumps 126 years after windmills; printing 15 years after pumps; watches 37 years after printing; the spinning-wheel 53 years after watches; the steam-engine 119 years after the spinning-wheel; the fireengine 14 years after the steam-engine; the spinning-frame 98 years after the fire-engine; and the electro-magnetic telegraph 71 years after the spinning-frame. What year was the electromagnetic telegraph invented? Ans. 1832.

4. During the American Revolutionary war, the British lost at the battle of Lexington 273 men, at Bunker Hill 1054 men, at Long Island 400 men, at White Plains 300 men, at Fort Washington 1000 men, at Trenton 1020 men, at Princeton 400 men, at Hubbardton 200 men, at Bennington 800 men, at Brandywine 500 men, at Stillwater 600 men, at Germantown 500 men, at Saratoga 400 men, by Burgoyne's surrender 5791 men, at Fort Mercer 500 men, at Monmouth 400 men, at Rhode Island 260 men, at Brier Creek 23 men, at Stony Point 600 men, at Savannah 130 men, at Camden 325 men, at King's Mountain 1150 men, at Cowpens 800 men, at Guilford Court-House 600 men, at Hobkirk's Hill 250 men, at Eutaw Springs 700 men, at Yorktown 7000 men. What was the entire loss? Ans. 25976 men.

5. Required the number of square miles in the following States, there being in Maine 35,000, in New Hampshire 8,030, in Vermont 8,000, in Massachusetts 7,250, in Rhode Island 1,200, in Connecticut 4,750, in New York 46,000, in New Jersey 6,851, in Pennsylvania 47,000, in Ohio 39,964, in Michigan 56,243, in Indiana 33,809, in Illinois 55,405, in Wisconsin 53,924, in Iowa 50,914, and in California 188,982. Ans. 643,322.

6. Required the number of square miles in the following States, there being in Delaware 2,120, in Maryland 11,000, in Virginia 61,352, in North Carolina 45,500, in South Carolina 28,000, in Georgia 58,000, in Alabama 50,722, in Florida 59,268, in Mississippi 47,147, in Tennessee 44,000, in Kentucky 37,680, in Missouri 67,380, in Arkansas 52,198, in Louisiana 46,431, and in Texas 325,520. Ans. 936,318.

7. According to the census of 1850, Maine had 583,169

inhabitants, New Hampshire 317,976, Massachusetts 994,514, Rhode Island 147,545, Connecticut 370,792, Vermont 314,120, New York 3,097,394, New Jersey 489,555, Pennsylvania 2,311,786, Delaware 91,532, Maryland 583,034, District of Columbia 51,687, Virginia 1,421,661, North Carolina 869,039, South Carolina 668,507, Georgia 906,185, Florida 87,445, Alabama 771,623, Mississippi 606,526, Lousiana 517,762, Texas 212,592, Arkansas 209,897, Tennessee 1,002,717, Missouri 682,044, Kentucky 982,405, Ohio 1,980,329, Indiana 988,416, Illinois 851,470, Michigan 397,654, Wisconsin 305,391, Iowa 192,214, California 92,597, and the Territories 92,298. What was the whole number of inhabitants? Ans. 23,191,876.

SUBTRACTION.

49. SUBTRACTION is the process of taking one number from another to find the difference.

When the two numbers are unequal, the larger is called the Minuend and the less number the Subtrahend; and when the numbers are equal, either is the Minuend, and the other is the Subtrahend. The result, or the number found by the subtrac tion, is called the Difference, or Remainder.

One number can be subtracted from another only when the units of both are of the same kind; and when the numbers are simple, the process is termed Subtraction of Simple Numbers.

50. To subtract simple numbers. Ex. 1. From 935 take 673.

OPERATION.

Minuend 935
Subtrahend 67 3
Remainder 262

Ans. 262.

We first take the 3 units from the 5 units, and find the difference to be 2 units, which we write under the figure subtracted. We then proceed to take the 7 tens from the 3 tens above it; but we here find a difficulty, since the 7 is greater than the 3. and cannot be subtracted from it. We therefore add 10 tens to the 3 tens, which makes 13 tens, and then subtract the 7 from 13, and 6 tens remain, which we write below. Then, to compensate for the 10 tens, equal to 1 hundred, added to the 3 tens in the minuend, we add 1 hundred to the 6 hundred of the subtrahend, which makes

7 hundreds, and subtract the 7 hundreds from 9 hundreds, and 2 hundreds remain. By adding the 10 tens to the minuend and the 1 hundred to the subtrahend, the two numbers being equally increased, their difference is not changed. (Art. 24, Ax. 9.) The remainder is

262.

NOTE. The addition of 10 to the minuend is sometimes called borrowing 10, and the addition of 1 to the subtrahend is called carrying 1.

RULE.— Place the less number under the greater, so that units of the same order shall stand in the same column.

Commencing at the right hand, subtract each figure of the subtrahend from the figure above it.

If any figure of the subtrahend is larger than the figure above it in the minuend, add 10 to that figure of the minuend before subtracting, and then add 1 to the next figure of the subtrahend.

- Add the remainder and the

51. First Method of Proof. subtrahend together, and their sum will be equal to the minuend, if the work is right.

This method of proof depends on the principle, that the greater of any two numbers is equal to the less added to the difference between them.

52. Second Method of Proof. Subtract the remainder or difference from the minuend, and the result will be like the subtrahend, if the work is right.

This method of proof depends on the principle, that the smaller of any two numbers is equal to the remainder obtained by subtracting their difference from the greater.

EXAMPLES.

2.

2.

3.

3.

OPERATION.

OPERATION AND PROOF. OPERATION. OPERATION AND PROOF.

81715?

27.

Weeks.

10000000

9999999

28. What is the value of 6767851

29. What is the value of 761619161 - 916781?

361784?

100716?
5971621 ?

30. What is the value of 31671675 31. What is the value of 16781321 32. What is the value of 1002007000 33. Sir Isaac Newton was born in the he died in 1727; how old was he at the cease?

34. Gunpowder was invented in the year

year 1642, and time of his deAns. 85 years. 1330; how long

was this before the invention of printing, which was in

1440?

Ans. 110

years.

35. 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? Ans. 190 years.

36. What number is that, to which if 6956 be added, the sum will be one million? Ans. 993044. 37. 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.

38. Bonaparte was declared emperor in 1804; how many years since?

39. The union of the government of England and Scotland was in the year 1603; how long was it from this period to 1776, the time of the declaration of the independence of the United States? Ans. 173 years.

40. 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.

41. From the Creation to the Deluge was 1656 years; thence to the founding of Rome 1595 years; thence to the death of Charlemagne, which took place 814 years after Christ, 1567 years. In what year of the world was Christ born? Ans. 4004.

42. A gentleman 83 years old has two sons; the age of the older son added to his makes 128 years, and the age of the younger son is equal to the difference between the age of the father and that of the older son. How old is each of his sons? Ans. The older, 45 years; the younger, 38 years.

43. During the year 1810, there were manufactured in the United States one hundred and forty-six thousand nine hundred and seventy-four yards of cotton cloth; and during the year 1855, five hundred and twenty million yards. What was the increase? Ans. 519,853,026 yards.

53. Method of subtracting, when there are two or more subtrahends.

Ex. 1. From a pile of wheat containing 657 bushels, A is to have 141 bushels, B 244 bushels, C 134 bushels, and D the remainder. How bushels is D to have? Ans. 138. many