21. Hence, for the addition of simple numbers, we have this RULE I. Place UNITS under UNITS, TENS under TENS, &c. II. Add up the row of units, set down the unit figure of the sum, and carry ONE for every TEN to the next left hand row or row of tens.

III. Do the same with each row, observing to set down the whole sum of the last.

William has 54 cents, Charles 76, and Henry 59: how many cents have all three ? Ans. 189 cents. How many acres in 3 farms, the first containing 575, the second 599, and the third 514 acres?

Ans. 1688.

22. PROOF. Begin at the TOP and add downward.

Let the scholar now answer to the following questions. What is the sum of 2 and 4? 4 and 6? 6 and 8? 8 and 9? 3 and 5? 5 and 7? 7 and 8? 4 and 5? 5 and 6? 6 and 7? 7 and 8? 5 and 8? 6 and 9? 5 and 9? 3 and 8? 2 and 9? &c.*

23. This sign

is equal to.

=

signifies EQUALITY: and is read equals, or

As 3 and 5 = 8, that is to say, 3 and 5 equals 8, or is equal to 8.

24. This sign + signifies ADDITION, and is read and, plus,+

or more.

As 3+5=8, that is to say, 3 and 5, or 3 plus 5, or 3 more 5, is equal to 8.

EXAMPLES IN ADDITION.

1. A gentleman bought a chaise for 217 dollars, a horse for 96, and saddle for 15; how much did the whole come to? Ans. 328 dollars.

2. How much in 3 bags of money, one containing 756 dollars, another 817, and a third 989 ? Ans. 2562 dollars.

3. Bought 5 loads of hay, weighing as follows, viz. 1st load 2475 pounds, 2d 1964 pounds, 3d 1843 pounds, 4th 2276 pounds, 5th 1728 pounds; how much in all? Ans. 10286 pounds,

4. Money of gold and silver was first made at Argos 894 years before Christ; how long has it been in use?

5. Bought goods to the amount of 325 dollars; paid for truckage 2 dollars, freight 7 dollars, and my own expenses came to 19 dollars; what do they stand me in? Ans. 353 dollars. viz., 267 dollars,

6. Borrowed at different times, the following sums, 354 dollars, 75 dollars, 587 dollars; what is the amount?

Ans. 1283 dollars.

7. St. Paul's Cathedral, in London, cost 800,000 pounds; the Royal Exchange 80,000; the Mansion House 40,000; Black Friars' Bridge 152,840; Westminister Bridge 389,000; and the Monument 13,000. Required the amount of these sums? Ans. 1,474,840 pounds.

8. If one of the fore quarters of an ox weigh 176 pounds, the other 169: one of the hind quarters 185 pounds, the other 193; the hide 84 pounds. and tallow 78 pounds; what is the whole weight of the ox. Ans. 885 lb.

The teacher may extend these at pleasure.

A Latin word signifying more.

9. A gentleman left to his daughter, 14 thousand 14 hundred and 14 dollars; he left to his son 1735 dollars more than to his daughter; what was the son's portion? and what the whole estate?

Ans. {Son's portion 17,149 dollars. {Whole estate 32,563 ...

THEORETICAL QUESTIONS.

...

What are simple numbers? What is simple addition? What is the first part of the RULE? What the second?the third? Why do you carry one for every ten? Why is it that figures in combination have a tenfold relation to one another? What is the method of PROOF? What is the sign of EQUALITY? How is it read? What the sign of ADDITION? How is the sign of addition read? What is plus?

EXAMPLES FOR PRACTICE.

1. SIR ISAAC NEWTON was born in the year 1642, he lived to be 85 years of age; in what year did he die? Ans. 1727. 2. George Washington was born 1732, he lived to the age of 67; in what year of our Lord did he die. Ans. 1799.

3. Benjamin Franklin was born 1706, he lived to the age of 84; in what year did he die?

Ans. 1790.

4. Homer was born 733 years before Christ; how many years since? 5. Bought sugar to the amount of $327, Coffee $538, and tea $749; what is the amount?

Ans. $1614. 6. In the city of Pekin in China, is a bell weighing 120,000 pounds; at Nankin in the same empire, another of 50,000 lbs. ; and at Erfort in Upper Saxony, the great bell weighs 25,400 lbs. ; quere, the weight of the three. Ans. 195,400 lbs.

7. From the creation to the departure of the Israelites from Egypt was 2513 years; to the siege of Troy 307; to the building of Solomon's Temple 180; to the building of Rome 251; to the expulsion of the kings from Rome 244; to the destruction of Carthage 363; to the death of Julius Cæsar 102; to the Christian era 44; required the time from the Creation to the Christian era. Ans. 4004 years.

8. The population of the New England states in 1820 was as follows:Maine, 298,335; New Hampshire, 244,161; Vermont, 235,764; Massachusetts, 253,287; Rhode Island, 83,059; Connecticut, 275,248; what was the whole number of inhabitants in those states, at that time.

Ans. 1,389,854.

9. The distance from Portland, in the province of Maine, to Boston is 125 miles, from Boston to New-York 250, from New-York to Philadelphia 95, from Philadelphia to Baltimore 102, from Baltimore to Richmond in Virginia 176, from Richmond to Charleston, S. Carolina 540, and from Charleston, to Savannah, in Georgia 119; what is the distance from Portland to Savannah? Ans. 1507.

SUBTRACTION OF SIMPLE NUMBERS,

COMMONLY CALLED

SIMPLE SUBTRACTION.

25. SIMPLE SUBTRACTION is finding the DIFFERENCE between

TWO SIMPLE NUMBERS.

Illus. 1. William had 78 peaches, and Henry 36; how many had William more than Henry?

William had 78

Henry

"36

Here it is plain that we find the difference, for 6 (units) from 8 (units) evidently leaves 2 (units); and 3 tens from 7 tens as evidently leaves 4 tens: therefore 4 tens and 2 Difference 42 peaches. (units) comprise the whole difference 42, the answer. 2. After William had lost 3 of his peaches, how many had he then more than Henry?

Tens.

Here we cannot take 6 (the unit figure of the lower number) from 5 (the unit figure of the upper); yet it is nevertheless evident, that we can take 36 from 75. To do this, however, we shall have to separate 75, into its parts, 7 tens and 5 (units): or, borrowing (as it is called) 1 TEN from the 7, as seen below, and adding it to 3 9 peaches. 5 (the figure above), we separate it into 6 tens and 15

William had 75 left.
Henry
36
Difference

and

and

75=

36=

3

Units.

units. Then separating 36 into its parts, 3 tens and 6 units, we may begin and take 6 units from 15 units, by which we obtain a difference of 9 (units). Proceeding to the next left hand place, it is evident that we must either make the upper figure less by 1 (that we borrowed from it) or carry (as it is called) 1 to the next Difference lower figure, and take the sUM from the upper. It is plain that either of these two modes will give the same result: For 3 from 6 (the upper figure made less by 1) leaves 3, the same as 4 from 7. But the operation (6 from 15) may be made more easy; for we may evidently take the lower figure from the 1 TEN that we borrowed and add in the upper figure, saying, in this case, 6 from 10 leaves 4, and 5 (the figure above) is 9; set down 9 and carry 1, &c.

3

-9

26. From the illustrations now given, we deduce the following general RULE I. Place the less number under the greater, with UNITS under UNITS, TENS under TENS, &c.

II. Begin at the right hand, and take each figure in the lower line from the figure above :

III. But, if the figure in the lower line EXCEED that above it, take it from ten, and add the remainder to the upper figure; set the result underneath, CARRY 1 to the next lower figure, and proceed as before.

27. The upper number is called the MINUEND, the lower number the SUBTRAHEND, and the difference the REMAINDER.

28. This sign-signifies SUBTRACTION, and is read minus,* or less.

As 7-25, that is to say 7 minus 2, or 7 less 2, is equal to 5.

What is 9 minus 5 equal to? 7-4? 5-3? 8-5? 8-3? 9-4? 9-3? 9-22 7-5? 7-3? 7-2? 6-2 ? &c.

* A Latin word signifying less.

29. PROOF. Add the REMAINDER to the SUBTRAHEND, and if the work is right, THAT SUM will be equal to the MINUEND.

Apples. BASKET. From 7, the minuend.

Take

4, the subtrahend.
HAT.

Diff. 3, the remainder,
IN THE BASKET.
7 the minuend =

Proof

Edwin has 7 apples in a BASKET; he puts 4 of them in his HAT; how many remain in the BASKET?

If he put just 4 apples of the 7 in his HAT, it is very obvious, that he would leave 3 in the BASKET; and if he then put these 3 which remaine in the BASKET, also in his hat, he would obviously diminish the number of apples in the BASKET, to 0, and increase the number in his HAT to 7. But he would then have just as many apples in his HAT as he had before in the BASKET: Therefore, adding the REMAINDER to the SUBTRAHEND will give a sum equal to the MINUend.

the number in the HAT AGAIN.

And it is therefore also obvious that

30. Subtraction is the REVERSE of Addition.

William had 9 cents, he gave James 4; how many had he then?

9-4 how many

?

Ans. 5.

James had 5 cents, William gave him 4; how many had he then? 5+4

how many?

how many?

Ans. 9.

Ans. 5.

A man had 8 cows, he sold 3 of them; how many had he left?

8 3

A man sold 3 cows, and had 5 left; how many had he at first? 3+5 how many?

Ans. 8.

30a. Hence also it appears that addition may be PROVED by subtraction, as well as subtraction by addition. That is, by taking, successively, from the sum the SEVERAL NUMBERS which were added to make it ;-If the work is right there will be NO REMAINDER.

ADDITION.

SUBTRACTION, Proof.
15-7-5-3=0

9+7+5+ 324 sum, &c. 24-9-7 -5-3=0, &c.

7+5+3= 15 sum.

SUBTRACTION.

8-62 Diff.

7-34 Diff., &c.

30 b. And here it may be remarked, that EVERY OPERATION in Arithmetic has its OPPOSITE; and they are all founded upon the fundamental principle of INCREASE and DECREASE, of MORE and LESs, or of ADDITION and SUBTRACTION,

30 c. And it may be further remarked, that when any NUMBER or QUANTITY is operated upon by ANY RULE, and the result by its OPPOSITE to the same extent, that number or quantity will remain UNCHANGED.

10+515, and 15—5— 10.

12820, and 20-812. A man had 8 cows, he bought 5 more, how many had he then? 8+5 how many?

EXAMPLES IN SUBTRACTION.

1. Borrowed 817 dollars, paid 375; how much remains due?

Ans. 442 dollars. 2. A man bought 157 acres of land for 3617 dollars, of which he afterwards sold 79 acres for 2923 dollars; how much land has he left? and what does it stand him in?

Ans. He has 18 acres left, and it stands him in 694 dollars. 3. What is the difference between 717171, and 171717? Ans. 545454,

4. A man going into account with himself, finds his whole property amounts to 18406 dollars, and that he owes 10509 dollars; how does he stand? Ans. 7897 dollars clear property. 5. Bought goods to the amount of 7345 dollars, sold of the same, deducting profit, to the amount of 3579 dollars; what amount remains unsold? Ans. 3766 dollars.

6. The Arabic or Indian method of notation was brought into general use in Europe, about the year 970: how long since?

7. A merchant has to receive the following sums, viz., 819, 612, 705, 31, and 1756 dollars; he has to pay rent 150 dollars, and his note to the bank for 3789 dollars: Will he have money to borrow or to spare, and how much? Ans. He will have to borrow 16 dollars. 8. There are 365 days in a year, of which 52 are Sundays; how many working days in a year?

THEORETICAL QUESTIONS.

Ans. 313.

What is simple subtraction? What the first part of the RULE? the second? the third? What is the upper number called? the lower number?

the difference? What is the SIGN of subtraction? How is it read? What is minus? What is the method of PROOF? What is obvious from the illustration of this method? What also appears from the examples that follow? How? What, af ter this, do you find TO BE REMARKED? What FURTHER ?

EXAMPLES FOR PRACTICE.

1. Christopher Columbus landed in San Salvador 1492; King Charles was beheaded 1648, and his son restored in 1660; how many years between each of these events? Ans. 156, 12, and 168. 2. The mariner's compass was invented 1302, gunpowder 1344, and printing 1440; how many years between? Ans. 42, 96, and 138.

3. A man drew a TEN THOUSAND dollar prize, out of which he had to pay percentage 1500 dollars; how much had he left?

4. Two men set out from the same place at the same time; one goes 52 miles a day, and the other follows at the rate of 39 miles a day; how far will they be apart at the close of the first day?

of the third?

of the fourth?

of the second?

of the fifth?

Ans. to the last, 65 miles. 5. Three men bought a piece of land for 2356 dollars; the first man paid 1279 dollars, and the second 397 less than the first; how much did the second and third pay?

Ans. The second 882 dollars.
The third 195 dollars.

6. There are two numbers, the greater is 9268, and their difference 597; what is the less?

Ans. 8671,

7. A gentleman left, to his son and daughter, an estate of 32,563 dollars; he left to his son 17,149 dollars, and to his daughter the rest: What was the daughter's portion? Ans. $15,414. 8. If a farm and the buildings on it be valued at $10,000 and the buildings alone at $4567, what is the value of the land? Ans. $5433.

9. How much must you add to 358,642 to make 1,487,945?

Ans. 1,029,303.

10. Bought goods for $13,682, and sold them again for $15,293; did 1 gain or lose, and how much? Ans. I gained, $1611.

11. A man carried his produce to market; he sold his pork for $45, his cheese for $38, and his butter for $29; he received, in pay, salt to the value of $17, $10 worth of sugar, $5 worth of molasses, and the rest in money; how much money did he receive?

Ans. $80.