7. Find the sum of the numbers from 210 to 225 inclusive.

8. From 4541 to 4556 inclusive.

9. From 6773 to 6788 inclusive.

10. From 8805 to 8820 inclusive.

11. 100685 + 273+ 824 +9 +25= how many?`

12. 73+1427+876+ 493 + 62+6= how many?

+62

13. Add fifty-seven billions two hundred nine thousand and twenty; nine hundred five millions, nine hundred ninety thousand, eight hundred eighty-seven, and nine hundred forty-two billions ninety-three millions eight hundred thousand and ninety-three.

14. Find the sum of five hundred thirty-seven, twenty-six thousand four hundred eighty-five, seven million five hundred ninetyseven thousand three hundred eighteen, eight hundred sixty millions four hundred thousand two hundred; ninety billions eight millions six hundred thousand, and seven trillions nine hundred ninety-eight billions eight hundred seventy-seven millions six hundred sixty-five thousand five hundred forty-four.

15. If in 1850, Pennsylvania raised 15367691 bushels of wheat, Ohio 14487351 bushels, New York 13121498 bushels, Illinois 9414575 bushels, Indiana 6214458 bushels, Virginia 11212616 bushels, Michigan 4925889 bushels, and Maryland 4494680 bushels, how many bushels did all these States raise that year?

16. If the Caspian Sea contains 145123 square miles, Lake Superior 31534, Michigan 23155, Huron 23123, Erie 7799, Ontario 6901, Winnipeg 6495, and Great Salt Lake 1875, what is the area of all these bodies of water?

17. A began the year 1861 seventy-five thousand nine hundred and eighty-four dollars in debt, and at the beginning of 1865 he was worth, free of all debts, 864575 dollars. How much did he clear in those four years?

18. B began business with 49263 dollars, and ended by being 37877 dollars in debt after giving up all his property. How much did he lose?

19. A ship started from a place 2799 miles north of the Equator, and sailed due south to a place 2553 miles south of the Equator. How many miles did she sail?

20. A thermometer showed 37 degrees below zero one winter, and 102 degrees above zero the next summer. How much did it vary?

21. If Cyrus took Babylon 538 years before Christ, how many years ago was that event?

22. Find the sum of two hundred fifty-nine thousand six hundred thirty-eight, seventy-one thousand five, nineteen thousand ninety, six hundred four thousand two, three hundred twelve thousand eight hundred, fifty-seven thousand, twenty-five thousand sixty-six, and nine hundred three.

Ex. 23. Find the sum of 2345, 3761, 4983, and 5736, by adding more than one column at a time.

WRITTEN PROCESS.

2345
3761
4983

5736

16825

EXPLANATION.

If we take two columns at a time, we may proceed thus:-36 and 3 are 39, and 80 are 119, and 1 is 120, 60 are 180, and 5 are 185, and 40 are 225. Write the 25 and carry the 2 hundreds to 57 hundred, making 59 hundred; and 40 are 99, and 9 are 108, and 30 are 138, and 7 are 145, and 20 are 165, and 3 are 168 hundreds. Write the whole.

If we wished to add three columns at a time, we would proceed thus:-736 and 3 are 739, and 80 are 819, and 900 are 1719, and 1 is 1720, and 60 are 1780, and 700 are 2480, and 5 are 2485, and 40 are 2525, and 300 are 2825, write the 825, and carry the 2 thousands; &c.

34. A manufacturer's weekly expenditures for a year were, respectively, 739, 546, 800, 406, 628, 942, 1312, 825, 1536, 597, 1200, 1031, 304, 648, 846, 2037, 1991, 1404, 1655, 2577, 2332, 3454, 4286, 2970, 1888, 1613, 2700, 3682, 4550, 3113, 1077, 1165, 890, 980, 573, 753, 654, 1659, 1296, 1010, 1313, 2357, 125, 693, 400, 999, 2345, 3452, 2765, 1987, 1828, and 1525 dollars. What were his total expenditures that year?

Art. 34.* Numbers expressed in other scales of notation are added on the same plan as decimal numbers, viz.: carrying when the sum of any column is greater than the number of units in that order.

Ex. 35. What is the sum of three, two, and three, expressed in the binery notation? Ans. 1000. (Eight.)

WRITTEN PROCESS.

Three is 1 1
Two is 10
Three is 11

EXPLANATION.

The sum of the column of units is two, which, in the binary notation, is 1 two and 0, or 10. Write the 0 units under the units, and carry the 1 two to the column of twos, making its sum four twos, which equal 1 eight, O fours, 0 twos. Write a 0 under the column of twos, another under the fours, and 1 under the eights, thus making the answer 1 eight, 0 fours, Ó twos, 0 units.

Eight is 10 00

*If desired, the learner can be exercised in the addition of numbers in any supposed notation, as explained in the preceding chapter. Or this may be omitted as a mere generalization of theoretical, but not practical interest or value; or it may be deferred till the pupil has learned to reduce numbers from the decimal to other scales.

CHAPTER IV.

SUBTRACTION OF SIMPLE WHOLE NUMBERS.

Art. 35.

Subtraction is the process of finding the difference between two numbers.

The terms in subtraction are, minuend, subtrahend, and difference, or remainder.

The minuend is that number from which a number is taken.

The subtrahend is that number which is taken from the minuend.

The difference, or remainder, is the number which is left after subtraction.

Art. 36. The sign of subtraction is a short line placed in the line of writing, thus It is called minus. It indicates that the number following it is to be taken from the number before it. Thus, 8-5-3 is read eight minus five equals three, or, is equal to three.

Art. 37. The minuend, subtrahend, and difference must be LIKE numbers.

NOTE. In such expressions as "There are ten more horses than cows in the field," we merely consider these objects as far as they are of the same kind, that is, as animals.

Art. 38. The subtraction of numbers containing more than one order of units, is based upon the following principles:

I. The whole difference of two numbers is equal to the sum of the differences of their corresponding parts.

II. Units of any order in the subtrahend can be directly taken only from units of the same order in the minuend.

III. If both minuend and subtrahend be equally increased, the difference will not be changed.

Ex. 1. From 735 take 349.

WRITTEN PROCESS.

735
349

386

subtrahend by 10, in

EXPLANATION.

Since 9 units cannot be taken from 5 units, we add 10 units to 5 units, making 15 units; then 9 units from 15 units leave 6 units. Write 6 under the column of units. Since we increased the minuend by 10, we increase the order not to alter the difference. (Prin. III.) Therefore, we add 1 ten to the 4 tens of the subtrahend, making it 5 tens. Since 5 tens cannot be taken from 3 tens, we add 10 tens to the 3 tens, making 13 tens: then, 5 tens from 13 tens leave 8 tens. Write 8 under the column of tens. Since we increased the minuend by 10 tens, or 1 hundred, we increase the subtrahend as much by adding 1 hundred to the 3 hundreds, making 4 hundreds: then 4 hundreds from 7 hundreds, leave 3 hundreds, making the whole difference 386.

MENTAL AND RECITATION MODEL.

9 from 5 impossible; 9 from 15 leaves 6. Write 6, and carry 1 to 4, making 5: 5 from 3 impossible: 5 from 13 leaves 8. Write 8, and carry 1 to 3, making 4: 4 from 7 leaves 3. Write 3, making the whole remainder 386.

Rule. Write the subtrahend under the minuend, units under units, tens under tens, &c., and draw a line under them.

Beginning with units, subtract successively each figure from the figure above it, if possible, and write the difference below.

When a figure exceeds in value the figure above it, add 10 to the upper, from this sum subtract the lower, and add one to the next lower figure, and proceed with the result as before.

NOTE. The process of adding 10 to the upper figure is often called borrowing ten. That of adding one to the next lower figure is called carrying one.

METHODS OF PROOF.

I. Add the remainder to the subtrahend; the result should equal the minuend.

II. Subtract the remainder from the minuend; the result should equal the subtrahend.