first period? In the second place of the third period? In the third place of the fourth period? In the first place of the fifth period? In the third place of the sixth period? In the second place of the seventh period? In the third place of the third period? In the first place of the seventh period? In the second place of the fourth pe、 riod? In the first place of the sixth period?

6. In 6480921 how many tens, and what remains? Ans. 648092 tens, and 1 unit remaining. How many hundreds, and what remains ? Ans. 64809 hundreds, and 21 remaining. How many millions, and what remains? thousands? ten-thousands ? hundred-thousands?

; 26. The names of the periods employed to express numbers higher than Quintillions are, in their order from Quintillions, Sextillions, Septillions, Octillions, Nonillions, Decillions, Undecillions, Duodecillions, Tredecillions, Quatuordecillions, Quindecillions, Sexdecillions, Septendecillions, Octodecillions, Novendeillions, Vigintillions, etc.

27. To read numbers, observe the following

RULE. Beginning at the units' place, point off the expression into periods of three figures each; then begin at the left, and read each period in order from left to right, giving after each, excepting he last, the name of the period.

29. Name the terms in the first example above, commencing with units (Art. 20, Remark). Ans. One unit, six tens, three hundreds. Name the terms in the second example. In the third. In the other examples, in their order.

Read from the Table (Art. 21), the number represented by the first six figures from the decimal point; the first eight; the first ten; nine; twelve; fifteen; seventeen; twenty; fourteen; eighteen.

30. To write numbers, observe the following

RULE. Beginning with the highest period, write the figures of each period in their order from left to right, filling vacant places with zeros.

Write the following:

one.

31. EXERCISES.

1. Three hundred sixty-four.

2. Seven thousand eighty-nine. 3. Eighteen thousand eighteen.

4. Nine hundred thousand sixteen.

Ans. 364.

Ans. 7089.

5. Four hundred twenty thousand, six hundred eighty-three.

6. Eight hundred ten thousand, two hundred four.

7. Two hundred fifty-nine thousand, seventy.

8 Forty-five million, seven hundred thousand, two hundred fifty

9. Nine hundred one million, two hundred eighteen thousand, twenty-two.

10. Three billion, thirty-seven million, nine hundred six thousand, two hundred.

11. Two hundred thirty-four million, eight hundred sixty-three thousand, three hundred eighty-nine.

12. Seventeen billion, seven hundred fifty-nine million, ninety thousand, sixty-seven.

13. Three hundred thirty-three quadrillion, seven hundred seventynine billion, three hundred thousand, two.

14. Nine hundred ten quadrillion, four million, three thousand. 15. Fifty-four quintillion, eighty-three quadrillion, nine hundred million, seventeen thousand, one hundred eighty-two.

16. Eighteen billion, four.

17. Forty million, eight hundred thousand.

18. Eighty-nine million, four hundred five thousand, seven. 19. Thirty-seven trillion, ninety-three billion, eighty-one. 20. Seven hundred quintillion, one quadrillion, one.

21. Fifty quintillion, forty-nine thousand, thirty.

32. We have seen that the value represented by a figure increases by a scale of tens, as the figure is removed towards the left, and decreases in the same manner as it is removed towards the right.

Applying this principle, we can represent parts of units by placing figures at the right of the decimal point.

If we consider a unit to be composed of ten equal parts, we may represent one or more of these parts, which are called tenths, by a figure in the first place at the right of the point; again, if we consider one of these tenths to be composed of ten equal parts, we may represent one or more of these parts, which are called hundredths, by a figure in the second place, and so on.

The first place at the right of the point is the tenths' place, the second, the hundredths' place, the third, the thousandths' place.

Here the 7 at the right of the point represents seven tenths of a whole one, the 8 represents eight hundredths, and the 5 rep. resents five thousandths. The entire number is read seven hun ired eighty-five thousandths; .25 is read twenty-five hundredths; 1 is read three tenths.

ADDITION.

33. Addition is the process of finding a number equal in value to two or more given numbers of the same kind. The number thus obtained is called the sum, or amount.

An upright cross, +, read plus, is the sign of addition, and, placed between two numbers, signifies that the one is to be added to the other. Two horizontal lines, =, read equal to, are the sign of equality, and signify that the quantities between which they are placed, are equal; thus, 2 + 5 = 7, is read, two plus five is equal to seven, or, two plus five equals seven.

ILLUSTRATIVE EXAMPLE.

34. Add the numbers 321, 285, and 937.

OPERATION.

321

285

937

Ans. 1543

We first write these numbers, units under units, tens under tens, hundreds under hundreds, and draw a line beneath. Then, adding the units first, 7 + 5+ 1= 13 units=1 ten and 3 units; we write the 3 in the units' place, under the column of units, and reserve the 1 ten to add with the column of tens. 1 ten+3 tens + 8 tens + 2 tens = 14 tens= 1 huncred and 4 tens; we write the 4 tens in the tens' place, and reserve the 1 hundred to add with the column of hundreds. 1 hundred +9 hundreds +2 hundreds +3 hundreds 15 hundreds 1 thousand and 5 hundreds; we write the 5 hundreds in the hundreds' place, and the 1 thousand in the thousands' place, and thus find the amount of the given numbers to be one thousand five hundred forty-three. Hence we derive the following

RULE FOR ADDITION. Write the numbers, units under units. tens under tens, hundreds under hundreds, etc. Begin to add ar the units' column. If the sum of the units is less than ten, write it under the column of units; if ten, or a number greater than ter, place the units' figure under the column of units, and reserve the tens to add with the tens. Proceed in the same way with the other columns, writing down the entire amount of the last column.

PROOF I. Add each column in a reverse direction; if ths same result be obtained as before, the work may be presumed to be

correct.

NOTE. - Greater readiness will be attained by mentioning only the results in adding columns. Thus, in the above example, instead of saying 7 and 5 are 12, and 1 are 13, say 7, 12, 13; and instead of saying, 1 ten and 3 tens are 4 tens, and 8 tens are 12 tens, and 2 tens are 14 tens, say 1, 4, 12, 14 tens.

35. EXAMPLES FOR PRACTICE.

1. What is the sum of twenty-one, sixty-seven, eighty-nine, thirty-two, forty-five, thirteen, ninety, and seventy-eight?

Ans. 435. 2. What is the sum of six hundred four, nine hundred ninetynine, seven hundred ten, six thousand nine hundred eighty-two, eleven thousand eight hundred seven ? Ans. 21,102.

3. What is the sum of 326, 981, 362, 707, 889, and 864?

Ans. 4129.

4. What is the sum of 246, 368, 909, 896, 763, and 892?

Ans. 4074.

5. What is the sum of 32689, 86543, 94861, 18325, and 90026? Ans. 322,444.

6. What is the sum of all the numbers from one to thirty, inclusive? Ans. 465. 7. What is the sum of all the numbers from one hundred fifty to one hundred seventy-five, inclusive?

8. Add 99, 364, 77, 86, 912, 32678, 96542, and 32684. 9. Add 987, 5, 679, 369, 153, 888, 806, 17, 27, and 5654. 10. Add 915, 875, 617, 868, 575, 387, 694, 946, and 6377. 11. Find the sum of the last four answers. Ans. 189,506. 12. Add 987, 425, 672, 307, 216, 321, 111, 872, 564, 876, 818. 419, 187, 160, and 3453.

13. 875 +466 + 327 + 942 + 286 + 424 + 309 + 429 +482317+406 +466 +111+171 +1618 what? 14. 324 + 868 + 522 + 297 + 789 + 524 + 286 + 361 +472 + 884 + 472 + 287 + 649 +592 +1788 what? 15. 876205 + 918 + 468 + 207 + 948 + 572 +618 +861+594 +872 +206 +48 +500+918+1331=what?