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4. Nine units of the 4th order, two of the 3d, four of the 1st. 5. Five units of the 4th order, eight of the 2d. 6. Five units of the 5th order, one of the 3d, eight of the 1st.

7. Three units of the 5th order, six of the 4th, four of the 3d, seven of the 1st.

8. Two units of the 6th order, four of the 5th, nine of the 4th, three of the 3d, five of the 1st.

9. Three units of the 8th order, five of the 7th, four of the 6th, three of the 5th, eight of the 4th, five of the 3d, eight of the 2d, seven of the 1st.

10. Three units of the 9th order, eight of the 7th, four of the 6th, six of the 5th, nine of the 1st.

11. Five units of the 12th order, three of the 11th, six of the 10th.

12. Four units of the 12th order, five of the 10th, eight of the 5th, nine of the 4th, four of the 3d.

13. Three units of the 15th order, six of the 14th, five of the 13th, three of the 9th, six of the 8th, five of the 7th, three of the 3d, six of the 2d, five of the 1st.

14. Five units of the 18th order, three of the 17th, six of the 16th, four of the 15th, seven of the 14th, eight of the 13th, four of the 12th, five of the 11th, six of the 10th, seven of the 9th, eight of the 8th, nine of the 7th, five of the 6th, six of the 5th, three of the 4th, two of the 3d, four of the 2d, eight of the 1st.

15. Two units of the 20th order, seven of the 19th, four of the 18th, eight of the 13th, five of the 6th, five of the 5th, five of the 4th, nine of the 1st.

Write the following numbers in figures :
16. Forty-eight.
17. One hundred sixty-four.
18. Forty-eight thousand seven hundred eighty-nine.

19. Five hundred thirty-six million three hundred forty-seven thousand nine hundred seventy-two.

20. Ninety-nine billion thirty-seven thousand four.

21. Eight hundred sixty-four billion five hundred thirty-eight million two hundred seventeen thousand nine hundred fifty-three.

22. One hundred seventeen quadrillion two hundred thirty-five trillion one hundred four billion seven hundred fifty million sixtysix thousand ten.

23. Ninety-nine quintillion seven hundred forty-one trillion fifty-four billion one hundred eleven million one hundred one.

24. One hundred octillion one hundred septillion one hundred quintillion one hundred quadrillion one hundred trillion one hundred billion one hundred million one hundred thousand one hundred.

25. Four decillion seventy-five nonillion three octillion fiftytwo septillion one sextillion four hundred seventeen quintillion ten quadrillion twelve trillion fourteen billion three hundred sixty million twenty-two thousand five hundred nineteen.

Write the following numbers in figures, and read them:

26. Twenty-five units in the 2d period, four hundred ninety-six in the 1st.

Ans. 25,496. 27. Three hundred sixty-four units in the 3d period, seven hundred fifteen in the 2d, eight hundred thirty-two in the 1st.

* 28. Four hundred thirty-six units in the 4th period, twelve in the 3d, one hundred in the 2d, three hundred one in the 1st.

29. Eighty-one units in the 5th period, two hundred nineteen in the 4th, fifty-six in the 2d.

30. Nine hundred forty-five units in the 7th period, eighteen in the 5th, one hundred three in the 3d.

31. One unit in the 10th period, five hundred thirty-six in the 9th, two hundred forty-seven in the 8th, nine hundred twenty-four in the 7th.

Point off and read the following numbers:
32.
564. 37.

2005. 33. 24835. 38.

100103. 34. 2474783. 39.

53000008. 35. 247843112. 40.

1001005003. 36. 23678542789. 41.

750000040003. 42. 247364582327896438542721.

43. 379403270506038009503070. 44. 20005700032004673000430512500000567304705030040.

23

ADDITION. 62. Addition is the process of uniting several numbers of the same kind into one equivalent number.

63. The Sum or Amount is the result obtained by the process of addition.

64. When the given numbers contain several orders of units, the method of addition is based upon the following principles :

I. If the like orders of units be added separately, the sum of all the results must be equal to the entire sum of the given numbers. (Ax. 10).

II. If the sum of the units of any order contain units of a higher order, these higher units must be combined with units of like order. Hence,

III. The work must commence with the lowest unit, in order to combine the partial sums in a single expression, at one operation. 1. Find the sum of 397, 476, and 873.

ANALYSIS. We arrange the numbers so that 397

units of like order shall stand in the same column. 476 We then add the first, or right hand column, and 873 find the sum to be 16 units, or 1 ten and 6 units ; 1746 writing the 6 units under the column of units, we

add the 1 ten to the column of tens, and find the sum to be 24 tens, or 2 hundreds and 4 tens; writing the 4 tens under the column of tens, we add the 2 hundreds to the column of hundreds, and find the sum to be 17 hundreds, or 1 thousand and 7 hundreds; writing the 7 hundreds under the column of hundreds, and the 1 in thousands' place, we have the entire sam, 1746.

65. From these principles we deduce the following

RULE. I. Write the numbers to be added so that all the units of the same order shall stand in the same column; that is, units under units, tens under tens, etc.

II. Commencing at units, add each column separately, and write the sum underneath, if it be less than ten.

OPERATION

III. If the sum of any column be ten or more than ten, write the unit figure only, and add the ten or tens to the next column.

IV. Write the entire sum of the last column. Notes.-1. In adding, learn to pronounce the partial results without naming the figures separately. Thus, in the operation given for illustration, say 3, 9, 16; 8, 15, 24; 10, 14, 17.

2. When the sum of any column is greater than 9, the process of adding the tens to the next column is called carrying.

66. Proof. There are two principal methods of proving Addition.

1st. By varying the combinations.

Begin with the right hand or unit column, and add the figures in each column in an opposite direction from that in which they were first added; if the two results agree, the work is supposed to be right.

2d. By excess of 9’s.

67. This method depends upon the following properties of the number 9:

I. If a number be divided by 9, the remainder will be the same as when the sum of its digits is divided by 9. Therefore,

II. If several numbers be added, the excess of 9's in the sum must be equal to the excess of 9's in the sum of all the digits in the numbers.

1. Add 34852, 24784, and 72456, and prove the work by the excess of 9's.

OPERATION.

sum

34852
24784
72456 ... 8, excess of 9's in all the digits of the numbers.

132092 ... 8, ANALYSIS. Commencing with the first number, at the left hand, we say 3 and 4 are 7, and 8 are 15; dropping 9, the excess is 6, which added to 5, the next digit, makes 11; dropping 9, the excess is 2; then 2 and 2 are 4, and 2 (the left hand digit of the second number) are 6, and 4 are 10; dropping 9, the excess is 1. Proceeding in like manner through all the digits, the final excess is 8; and as 8 is also the excess of 9's in the sum, the work of addition is correct. It is evident that the same result will be obtained by adding the digits in columns as in rows. Hence, to prove Addition by excess of 9's:

* For a demonstration of these properties, see 136, IX.

Commencing at any figure, add the digits of the given numbers in any order, dropping 9 as often as the amount exceeds 9. If the final excess be equal to the excess of 9's in the sum, the work is right

Note.—This method of proving addition by the excess of 9's, fails in the fol. lowing cases : 1st, when the figures of the answer are misplaced; 2d, when the value of one figure is as much too great as that of another is too small.

EXAMPLES FOR PRACTICE
(1.)
(2.)
(3.)

(4.)
8635 1234567

67

603
2194
723456

123 298765
7421
34565

4567 47321
5063
45666

89093 58653
2196

333
654321

5376
1245
90 1234567

340 26754 2038677 5. 123+456+785+12+345+901+567=how many ? 6. 12345+67890+8763+347 +1037+198760=how many ? 7. 172+4005+3761+20472+367012+19762=how many?

8. What is he sum of thirty-seven thousand six, four hundred twenty-nine thousand nine, and two millions thirty-six ?

Ans. 2,466,051. 9. Add eight hundred fifty-six thousand nine hundred thirtythree, one million nine hundred seventy-six thousand eight hundred fifty-nine, two hundred three millions eight hundred ninetyfive thousand seven hundred fifty-two.

Ans. 206,729,544. 10. What is the sum of one hundred sixty-seven thousand, three hundred sixty-seven thoasand, nine hundred six thousand, two hundred forty-seven thousand, seventeen thousand, one hundred six thousand three hundred, forty thousand forty-nine, ten thousand four hundred one ?

Ans. 1,860,750. 11. What number of square miles in New England, there

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