many ? figure only under the column ; the new ite next his lo the left hand 16. A man has three farms, one containing 500 acres, another 213 acres, and another 76 acres; how many acres in the three farms? 500 + 213 + 76 = how many? 17. Bought a farın for 2316 dollars, and afterwards sold it so as to gain 550 dollars; what did I sell the farm for? 2316 + 550 = how Hitherto the amount of any one column, when added up, has not exceeded 9; consequently has been expressed by a single figure. But it will frequently happen that the amount of a single column will excecd 9, requiring two or inore figures to express it. 18. There are three bags of money. The first contains 876 dol. lars, the second 653 dollars, the third 524 dollars; what is lue aniount contained in all the bags ? OPERATION. Writing down the numbers as already di. First bag, 876 rected, we begin with the right hand, or unit Second bag, 653 coluinn, and find the amount to be 13, that is, Third bag, 524 3 units and 1 ten. Setting down the three units, or right hand figure, in unit's place, diAmount, 2053 rectly under the colunin, we reserve the I ten, or left hand figure, to be added with the other tens, in the next column, saying, 1, which we reserved, to two, inakes 3, and 5 are 8, and 7 are 15, which is 5 units of its own order, and i unit of the next higher order, that is, 5 tens and i hundred. Setting down the 5 tens, or right hand figure, directly under the column of tens, we reserve the left hand figure, or i hundred, to lie added in the column of hundreds, saying, i lo 5 is 6, and 6 are 12, and 8 are 20, which being the last column, we set dowu the whole number, writing the 0, or right hand figure, directly under the column, and carry, ing forward the 2, or left hand figure, to the next place, or place of thousands. Wherefore, we find the whole amount of money coutained in the three bags to be 2053 dollars,-the answer. Proof. We may reverse the order, and, beginning at the top, add the figures downward. If the two results are alike, the work is supposed to be right. From the examples and illustrations now given, we derive the fol. lowing KULE. I. Write the numhers to be added, one under another, placing units under units, tens under tens, &c., and draw a line underneath. II. Begin at the right hand or unit column, and add together all the figures contained in that column ; if the amount does not exceed 9, write it under the column; but if the amount exceed 9, so that it shall require two or more figures to express it, write down the unit are higher order, which, being reserved, must be carried forward and added to the first figure in the next column. III. Add each succeeding column in the same manuer, and set down the whole amount at the last columy. EXAMPLES FOR PRACTICE. 19. A man bought four loads of hay; one load weighed 1817 pounds, another weighed 1950 pounds, another 2156 pounds, and another 2210 pounds; what was the amount of hay purchased? 20. A person owes A 100 dollars, B 2160 dollars, C 785 dollars, D 92 dollars; what is the amount of his debts ? 21. A farmer raised in one year 1200 hushels of wheat, 850 bushels of Indian corn, 1000 bushels of oats, 1086 bushels of barley, and 74 bushels of pease ; what was the whole amount ? Ans, 4210. 22. St. Paul's Cathedral, in London, cost 800,000 pounds sterling, the Royal Exchange 80,000 pounds, the Mansion House 40,000 pounds, Black Friars Bridge 152,840 pounds, Westminster Bridge 339,000 pounds, and the Nonument 13,000 pounds; what is the anowit of these sums ? Ans. 1,474,840 pounds. 23. At the census of 1820, the number of inhabitants in the New England states was as follows:-Maine 298,335, New Hampshire 244,161, Vermont 235,764, Massachusetts 523,287, Rhode Island 83,039, Connecticut 275,248; what was the whole number of inhabitants at that time in those states ? Ans. 1,659,854. 24. From the creation to the departure of the Israelites from Egypt was 2513 years ; to the siege of Troy, 307 years more; to the building of Solomon's Temple, 180 years; to the building of Rome, 251 years ; to the expulsion of the kings from Rome, 244 years; to the destruction of Carthage, 363 years ; to the death of Julius Cæsar, 102 years; to the Christian era, 44 years; required the time from the creation to the Christian era. Ans. 4004 years. 29. What is the amount of 46723, 6742, and 986 dollars ? 30. A man has three orchards; in the first there are 1.10 trees that bear apple's, and 64 trees that hear peaches; in the second, 234 trees bear apples, and 73 bear cherries ; in the third, 47 trees bear plums, 36 bear pears, and 25 bear cherries ; how many trees in all the orchards? SUPPLEMENT TO NUMERATION AND ADDITION. QUESTIONS. 1. What is a single or individual thing called ? 2. What is nota. tion ? 3. What are the methods of potation uow in use ? “4. How many are the Arabic characters or figures ? 5. What is numeration i 6. What is a fundamental law in notation? 7. What is addition : 8. What is the rule for adulition ? 9. What is the result, or nuniler sought, called ? 10. What is the sign of addition ? 11. ot equality? 12. How is addition proved ? EXERCISES. 1. Washington was horn in the year of our Lord 1732; he was 67 years old when he died; in what year of our Lord did he die? 2. The invasion of Greece hy Xerxes took place 481 years before Christ ; how long ago is that this current year 1827 ? 3. There are two num!iers, the less number is 8671, the difference between the numbers is 597 ; what is the greater number? 4. A man borrowed a sum of money, and paid in part 684 dollars ; the suint left unpaid was 876 dollars; what was the sun horrowed ? 5. There are four numbers, the first 317, the second 812, the third 1350, and the fourth as much as the other three; what is the sun of them all ? 6. A gentleman left his daughter 16 thousand, 16 hundred and 16 dollars; he left his son 1800 more than his daughter; what was his son's portion, and what was the amount of the whole estate ? Ans. Son's portion, 19,416. Whole estate, 37,0:32, 7. A man, at his death, left his estate to his four children, who after paying debts to the amount of 1476 dollars, received 4768 dollars each; how much was the whole estate ? Ans. 200548. 8. A man bought four hogs, each weighing 375 pounds; how much lid they all weigh? Ans, 1300. 7. The fore quarters of an ox weigh one hundred and eight pounds each, the hind quarters weigh one hundred and twenty-four poun:ls each, the hide seventy-six pounds, and the tallow sixty pounds; what is the whole weight of the ox? Ans. 600, 10. A man, being asked his age, said he was thirty-four years old when his ellest son was born, who was then fifteen years of age; what was the age of the father? 11. A man sold two cows for sixteen dollars each, twenty bushels of corn for twelve dollars, and one hundred pounds of tallow for eight dollars; what was his due ? SUBTRACTION 46. 1. Charles, having 19 cents, bought a book, for which ho gave o cents; how many cents had he left ? 2. John had 12 apples; he gave 5 of them to his brother; how many had he left ? 3. Peter played at marbles; he had 23 when he began, but when he had done he had only 12; how many did he lose ? 4. A man bought a cow for 17 dollars, and sold her again for 22 dollars; how many dollars did he gain ? 5. Charles is 9 years old, and Andrew is 13; what is the difference in their ages? 6. A iran borrowed 50 dollars, and paid all but 18; how inany dollars fiu he pay? that is, take 18 from 50, and how many would there lie left? 7. John bought a hook and slate for 33 cents; he gave 8 cents for the book; what did the slate cost him? 8. Peter bought a wagon for 36 cents, and sold it for 45 cents; how many cents did he gain by the bargain ? 9. Peter sold a wagon for 45 cents, which was 9 cents more than he gave for it; how many cents did he give for the wagon ? 10. A boy, being asked how old he was, said that he was 25 years younger than his father, whose age was 33 years; how old was the boy? The taking of a less number from a greater (as in the foregoing examples) is called Subtraction. The greater numher is called the minuend, the less number the subtrahend, and what is left after subtraction is called the difference, or remainder. 11. If the minuend be 8, and the subtrahend 3, what is the difference or reinainder? Ans. 5. 12. If the subtrahend be 4, and the minuend 16, what is the remainder ? 13. Samuel bought a book for 20 cents; he paid down 12 cents; how many cents more must he pay? Sigx. A short horizontal line, 5, is the sign of subtraction. It is usually read minus, which is a Latin word signifying less. It shows that the number afler it is to be taken from the number before it. Thus, 8-3=5, is read 8 minus or less 3 is equal to 5; or, 3 from 8 leaves 5. The latter expression is to be used by the pupil in committing the following SUBTRACTION TABLE. 3 = how many ? 8 -5= how many? 9 -4 = how many ? 12 3 = how many? 13 -4 = how many ? 18 -7 = how many ? how inany? 17. When the numbers are small, as in the foregoing examples, the taking of a less number from a greater is readily done in the mind; but when the nurnhers are large, the operation is most easily performed part at a time, and therefore it is necessary to write the numbers down before performing the operation. 14. A farmer, having a Hock of 237 sheep, lost 114 of them by disease; how many had he left? Here we have 4 units to be taken from 7 units, 1 ten to be taken from 3 tens, and i hundred to be taken from 2 hundreds. It will therefore be most convenient to write the less number under the greater, observing, as in addition, to place units under units, lens under tens, &c., thus : We now hegin with the units, OPERATION. From 237 the minuend, saying, 4 (units) froni ?, (units,) Take 114 the subtrahend, and there remain 3, (units wluch we set down directly under the column in unit's place. Then, pro123 the remainder. ceeding to the next column, we say, 1 (ten) from 3, (tens,) and there remain 2, (tens,) which we set down in ten's place. Proceeding to the next column, we say, ! (hundred) from 2, (hundreds,) and there remains !, (hundred,) which we set down in hundred's place, and the work is done. It now appears, that the nuinber of sheep left was 123; that is, 237 - 114=123. After the same manner are performed the following examples: 15. There are two farms; one is valued at 3750, and the other at 1500 dollars; what is the difference in the value of ihe two farms? 16. A man's property is worth 8560 dollars, but he has dehts to the amount of 3500 dollars; what will remain after paying his debts? 17. James, having 15 cents, bought a penknife, for which he gave 7 cents; how inany cents had he left ? OPERATION. 15 cents. A difficulty presents itselfhere ; for we connot 7 cents. take 7 from 5; hul we can lake 7 uomi 15, and there will remain 8. 8 cents left. 18. A man lought a horse for 85 dollars aui a cow for 27 dollars ; what did the horse cost hum more than the cow ? OPERATION. The same dificult' meels us here as in thic Horse, 85 last example; we cannot take 7 from 5 ; hiul in Cow, 27 the last examp!- the larger mumler consisted of i ten and 5 units, which together make 15 ; we Difference, 58 therefore took 7 from 15. Here we have 8 tels |