Charles Davies.

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Font size 9. Write 60 units of the third order, with four of the 2d,
and 5 of the first.

10. Write 6 units of the 4th order, with 8 of the 3d,
4 of the 1st.

25. To what are ten units of the third order equal ? How do you
write it? How is a single unit of the first order written ? How do
you write a unit of the second order ? One of the third ? One of the
fourth ? One of the fifth ?

26. On what does the unit of a figure depend ? What is the unit of
the first place on the right ? What is the unit of the second place ?
What is the unit of the third place ? Of the fourth ? Of the fifth ?
Sixth ? How many units of the first order make one of the second ?
How many of the second one of the third ? How many of the third one
of the fourth, &c. When figures are written by the side of each other,
how many units of any place make one unit of the place next to the
left?

NUMERATION. 17

11. Write 9 units of the 5th order, of the 4th, 8 of the
3d, 1 of the 2d, and 3 of the 1st.

12. Write 7 units of the 6th order, 8 of the 5th, of the
4th, 5 of the 3d, 7 of the 2d, and 1 of the llth.

13. Write 9 units of the 7th order, of the 6th, 2 of the
5th, 3 of the 4th, 9 of the 3d, 2 of the 2d, and 9 of the 1st.

14. Write 8 units of the 8th order, 6 of the 7th, 9 of the
6th, 8 of the 5th, 1 of the 4th, of the 3d, 2 of the 2d, and
8 of the 1st.

15. Write 1 unit of the 9th order, 6 of the 8th, 9 of the
7th, 7 of the 6th, 6 of the 5th, 5 of the 4th, 4 of the 3d, 3 of
the 2d, and 2 of the 1st.

16. Write 8 units of the 10th order, of the 9th, of the
8th, of the 7th, 9 of the 6th, 8of the 5th, of the 4th,
3 of the 3d, 2 of the 2d, and of the 1st.

17. Write 7 units of the ninth order, with 6 of the 7th, 9
of the third, 8 of the 2d, and 9 of the 1st.

18. Write 6 units of 8th order, with 9 of the 6th, 4 of the
5th, 2 of the 3d, and 1 of the 1st.

19. Write 14 units of the 12th order, with 9 of the 10th,
6 of the 8th, 7 of the 6th, 6 of the 5th, 5 of the 3d, and 3
of the first.

20. Write 13 units of the 13th order, 8 of the 12th, 7 of
the 9th, 6 of the 8th, 9 of the 7th, 7 of the 6th, 3 of the 4th,
and 9 of the first.

21. Write 9 units of the 18th order, 7 of the 16th, 4 of the
loth, 8 of the 12th, 3 of the llth, 2 of the 10th, 1 of the 9th,
of the 8th, 6 of the 7th, 2 of the third, and 1 of the 1st.

NUMERATION.

27. NUMERATION is the art of reading correctly any num-
ber expressed by figures or letters.

one to one thousand. The Numeration Table will teach him
to read any number whatever ; or, to express numbers in words.

27. What is Numeration? What is the unit of the first period?
What is the unit of the second ? Of the third ? Of the fourth ? Of
the fifth? Sixth? Seventh? Eighth? Give the rale for reading
numbers.

NUMERATION.

NUMERATION TABLE.

6th Period, 5th Period. 4th Period. 3d Period, 2d Period. 1st Period.
Quadrillions. Trillions. Billions. Millions. Thousands. Units.

II; I ! ! I ! ! I ! ! l-s :

ip . ?. * ! ^ 8 -^1 i

S3 -25 ||| ||| |a| |

,

6,
8 2,

6,
7 5,
879,
023,
301,

,

.

.

123,

087,

7,

000,

735,

B

.

.

4 3,

2 1 0,

460,

548,

000,

087,

( .

.

6,

245,

289,

421,

7 2,

549,

1 3 6,

822,

894,

602,

043,

288,

7,

641,

000,

907,

456,

8 4,

912,

876,

4 1 9,

285,

912,

761,

257,

327,

826,

6,

407,

2 1 2,

936,

876,

541,

5 7,

289,

678,

541,

297,

313,

920,

323,

842,

768,

319,

675,

NOTES. 1. Numbers expressed by more than three figures are
written and read by periods, as shown in the above table.

2. Each period always contains three figures, except the last,
which may contain either one, two, or three figures.

3. The unit of the first, or right-hand period, is 1 ; of the second
period, 1 thousand ; of the 3d, 1 million ; of the fourth, 1 billion ;
and so, for periods, still to the left.

4. To quadrillions succeed quintillions, sextillions, septillions,
octillions, &c.

5. The pupil should be required to commit, thoroughly, the
names of the periods, so as to repeat them in their regular order
from left to right, as well as from right to left.

NUMERATION.

19

I. Divide the number into periods of three figures each,
beginning at the right hand.

II. Name the order of each figure, beginning at the right
hand.

III. Then, beginning at the left hand, read each period an
if it stood alone, naming its unit.

28. Let the pupil point off and read the following numbers
-then write them in words.

19.
20.
21.
22.

67

125

6256

4697

23697

412304

7.

8.

9.
10.
11.
12.

6124076
8073405
26940123
9602316
87000032
1987004086

13.

14.
15.
16.
17.

18.

804321049
90067236708
870432697082
1704291672301
3409672103604
49701342641714

8760218760541

904326170365

30267821040291

907620380467026

23. 9080620359704567

24. 9806071234560078

25. 30621890367081263

26. 350673123051672607

NOTE. Let each of the above examples, after being written on
the black board, be analyzed as a class exercise ; thus :

Ex. 1. How many tens in 67 ? How many units over ?

2. In 125, how many hundreds in the hundreds place? How
many tens in the tens place ? How many units in the units
place ? How many tens in the number ?

3. In 6256, how many thousands in the thousands place ? How
many hundreds in the hundreds place ? How many tens in the
tens place ? How many units in the units place ?

4. How many thousands in the number 4697? How many
hundreds ? How many tens ? How many units ?

5. How many thousands in the number 23697? How many
hundreds ? How many tens ? How many units ?

6. How many hundreds of thousands in 412304? How many
ten thousands ? How many thousands ? How many hundreds ?
How many tens ? How many units ?

28. Name the units of each order in example 9 ? In 10 ? In 15 ?
In 30 ? Give the rule for writing numbers.

20 NUMERATION.

RULE FOR WRITING NUMBERS, OR NOTATION.

I. Begin at the left hand and write each period in order, as
if it icere a period of units.

II. When the number of any period, except the left hand
period, is expressed by less than three figures, prefix one or two
ciphers ; and when a vacant period occurs, fill it with ciphers.

EXAMPLES IX NOTATION.

29. Express the following numbers in figures :

1. One hundred arid five.

2.i Three hundred and two.

3. Five hundred and nineteen.

_. 4. One thousand and four.

5. Eight thousand seven hundred and one.

6. Forty thousand four hundred and six. /

7. Fifty-eight thousand and sixty-one.

8. Ninety-nine thousand nine hundred and ninety-nine.

9. Four hundred and six thousand and forty-nine.

10. Six hundred and forty-one thousand, seven hundred
and twenty-one.

11. One million, four hundred and twenty-one thousands,
six hundred and two.

12. Nine millions, six hundred and twenty-one thousands,
and sixteen. / ~j

13. Ninety-four millions, eight hundred and seven thous-
ands, four hundred and nine.

14. Four billions, three hundred and six thousands, nine
hundred and nine.

15. Forty-nine billions, nine hundred and forty-nine thous-
ands, and sixty-five.

16. Nine hundred and ninety billions, nine hundred and
ninety-nine millions, nine hundred and ninety thousands, nine
hundred and ninety-nine.

17. Four hundred and nine billions, two hundred and nine
thousands, one hundred and six.

18. Six hundred and forty-five billions, two hundred and
sixty-nine millions, eight hundred and fifty-nine thousands,
nine hundred and six.

NUMERATION. iJl

19. Forty-seven millions, two hundred and four thousands,
eight hundred and fifty-one.

20. Six quadrillions, forty-nine trillions, seventy-two bil-
lions, four hundred and seven thousands, eight hundred and
sixty-one.

21. Eight hundred and ninety-nine quadrillions, four hun-
dred and sixty trillions, eight hundred and fifty billions, two
hundred millions, five hundred and six thousands, four hun-
dred and ninety-nine.

22. Fifty-nine trillions, fifty-nine billions, fifty-nine millions,
fifty-nine thousands, nine hundred and fifty-nine.

23. Eleven thousands, eleven hundred and eleven.

24. Nine billions and sixty-five.

25. Write three* hundred and four trillions, one million,
three hundred and twentv-one thousands, nine hundred and
forty-one.

26. Write nine trillions, six hundred and forty billions,
with 7 units of the ninth order, 6 of the seventh order, 8 of
the fifth, 2 of the third, 1 of the second, and 3 of the first.

27. Write three hundred and five trillions, one hundred
and four billions, one million, with 4 units of the fifth order,
5 of the fourth, 7 of the second, and 4 of the first.

28. Write three hundred and one billions, six millions, four
thousands, with 8 units of the fourteenth order. 6 of the
third, and two of the second.

29. Write nine hundred and four trillions six hundred and
six, with 4 units of the eighteenth order, five of the sixteenth,
four of the twelfth, seven of the ninth, and 6 of the fifth.

30. Write sixty-seven quadrillions, six hundred and forty-
one billions, eight hundred and four millions, six hundred and
forty-four.

31. Write eight hundred and three quintillions, sixty-nine
billions, four hundred and forty millions, nine hundred thous-
and and three.

32. Write one hundred and fifty-nine sextillions, four hun-
dred and five billions, two hundred and one millions, three
thousand and six.

33. Write four hundred and four septillions, nine hundred
and three sextillions, two hundred and one quintillions, forty
quadrillions, and three hundred and four.

30. 1. John has two apples and Charles has three : how
many have both ?

ANALYSIS. If John's apples be placed with Charles's, there will
be five apples.

The operation of finding how many apples both have is called

2 and are 2

3 and are 3

4 and are 4

5 and are 5

2 and 1 are 3

3 and 1 are 4

4 and 1 are 5

5 and 1 are G

2 and 2 are 4

3 and 2 are 5

4 and 2 are G

5 and 2 are V

2 and 3 are 5

3 and 3 are G

4 and 3 are 7

5 and 3 are 8

2 and 4 are 6

3 and 4 are 7

4 and 4 are 8

5 and 4 are 9

2 and 5 are 7

3 and 5 are 8

4 and 5 are 9 5 and 5 are 10

2 and 6 are 8

3 and 6 are 9

4 and 6 are 10

5 and 6 are 1 1

2 and 7 are 9

3 and 7 are 10

4 and 7 are 11

5 and 7 are 12

2 and 8 are 10

3 and 8 are 11

4 and 8 are 12

5 and 8 are ]3

2 and 9 are 1 1

3 and 9 are 12

4 and 9 are 13

5 and 9 are 14

2 and 10 are 12

3 and 10 are 13

4 and 10 are 14

5 and 10 are 15

6 and are 6

7 and are 7

8 and are 8

9 and are 9

6 and 1 are 7

7 and 1 are 8

8 and 1 are 9

9 and 1 are 10

G and 2 are 8

7 and 2 are 9

8 and 2 are 10

9 and 2 are 11

G and 3 are 9

7 and 3 are 10

8 and 3 are 11

9 and 3 are 12

6 and 4 are 10

7 and 4 are 11

8 and 4 are 12

9 and 4 are 13

G and 5 are 11

7 and 5 are 12

8 and 5 are 13

9 and 5 are 14

6 and 6 are 12

7 and G are 13

8 and 6 are 14

9 and 6 are 15

6 and 7 are 13

7 and 7 are 14

8 and 7 are 15

9 and 7 are 16

G and 8 are 14

7 and 8 are 15

8 and 8 are 16

9 and 8 are 17

6 and 9 are 15

7 and 9 are 16

8 and 9 are 17

9 and 9 are 18

6 and 10 are 16

7 and 10 are 17

8 and 10 are 18

9 and 10 are 19

2. James has 5 marbles and William 7 ? how many have
both?

3. Mary has 6 pins and Jane 9 : how many have both ?

4. How many are 4 and 5 and 3 ?

5. How many are 6 and 4 and 9 ?

6. How many are 3 and 7 ? 4 and 6 ? 2 and 8 ? 5 and 5 ?
9 and 1? 10 arid ? and 10?

7. How many are 6 and 3 and 9 ? How many are 18 and
2? 18 and 3? 18 and 5?

SIMPLE NUMBERS. 23

8. James had 9 cents and Henry gave him eight more :
how many had he in all ?

PRINCIPLES AND EXAMPLES.

31. James has 3 apples and John 4 : how many have
both ? Seven is called the sum of the numbers 3 and 4.

The SUM of two or more numbers is a number which con-
tains as many units as all the numbers taken together.

ADDITION is the operation of finding the sum of two or
more numbers.

OF THE SIGNS.

32. The sign + is called plus, which signifies more.
When placed between two numbers it denotes that they are

The sign = is called the sign of equality. When placed
between two numbers it denotes that they are equal ;
that is, that they contain the same number of units. Thus :
3 + 2 = 5

2+3= how many?

1+2 + 4= how many ?

2 + 3 + 5 + 1= how many?

6 + 7+2+3= how many?

1 + 6 + 7+2 + 3= how many?

1+2+3+4 + 5 + 6 + 7+8 + 9= how many?

1. James has 14 cents, and John gives him 21 : how many
will he then have ?

OPERATION.

14

ANALYSIS. Having written the numbers, as at the 21
right of the page, draw a line beneath them.

oO cents.

The first number contains four units and 1 ten, the second 1
unit and two tens. We write the units in one column and the
tens in the column of tens.

31. What is the sum of two or more numbers? What is addition ?

32. What is the sign of addition ? What is it called ? What does
it signify? Express the sign of equality? When placed between two
numbers what does it show ? When is a number equal to the sum
of other numbers ? Give an example.

We then begin at the right hand, and say 1 and 4 are 5, which
we set down below the line in the units' place. We then add
the tens, and write the sum in the tens' place. Hence, the sum
is 3 tens and 5 units, or 35 cents.

OPERATION.

24

2. John has 24 cents, and William 62 : how 62
many have both of them ? gg

OPERATION.

3. A farmer has 160 sheep in one field, 20 in 1 ^
another, and 16 in another : how many has he

in all ?

196

OPERATION.

4. What is the sum of 328 and 111 ?

499

(5.) (6.) (7.) (8.)
427 329 3034 8094
242 260 6525 1602
330 100 236 103

999
9. What is the sum of 304 and 273 ?

10. What is the sum of 3607 and 4082 ?

11. What is the sum of 30704 arid 471912 ?

12. What is the sum of 398463 and 401536 ?

13. If a top costs 6 cents, a knife 25 cents, a slate 12
cents : what does the whole amount to ?

14. John gave 30 cents for a bunch of quills, 18 cents for
an inkstand, 25 cents for a quire of paper : what did the
whole cost him ?

15. If 2 cows cost 143 dollars, 5 horses 621 dollars, and 2
yoke of oxen 124 dollars : what will be the cost of them all *

16. Add 5 units, 6 tens, and 7 hundreds.

ANALYSTS. We set down the 5 units in the place oi
of units, the 6 tens in the place of tens, and the 7
hundreds in the place of hundreds. We then add up, "g ^ JS
and find the sum to be 765.

We must observe, that in all cases, units of the 5

same order are written in the same column. ^ 6

TT5"

SIMPLE NUMBERS. 25

1 7. What is the sum of 3 units, 8 tens, and 4 thousands ?

18. What is the sum of 8 hundreds, 4 tens, 6 units, and 6
thousands ?

19. What is the sum of 3 units, 5 units, 6 tens, 3 tens, 4
hundreds, 3 hundreds, 5 thousands, and 4 thousands?

20. What is the sum of five units of the 4th order, 1 of the
3d, three of the 4th, five of the 3d, and one of the 1st?

21. What is the sum of six units of the 2d order, five of the
3d, six of the 4th, three of the 2d, four of the 3d, two of the
1st, and four of the 2d?

22. What is the sum of 3 and 6, 5 tens and 2 tens, and 3
hundreds and 6 hundreds ?

23. What is the sum of 4 and 5, 5 tens, 3 hundreds and 2
hundreds ?

GENERAL METHOD.

33. 1. A farmer paid 898 dollars for one piece of land, and
637 dollars for another; how many dollars did
he pay for both ? OPERATION.

ANALYSIS. Write the numbers thus, 898

and draw a line beneath them.

sum of the units, - 15

sum of the tens, 12

sum of the hundreds, 1 4

sum total 1535

1. The example may be done in another way,

thus : Having set down the numbers, as before, OPERATION.

say, 7 units and 8 units are 15 units, equal to 898

1 ten and 5 units : set the 5 in the units' place, 63*7

and the 1 ten in the column of tens. Then n

say, 1 tea and 3 tens are 4 tens, and 9 tens are 1535
13 tens, equal to 1 hundred and 3 tens. Set

the 3 in the tens' place and the 1 hundred in the column of

33. How do you set down the numbers for addition ? Where do
you begin to add? If the sum of any column can be expressed by
a single figure, what do you do with it? When it cannot, what do
you write down ? What do you then add to the next column ? When
you add to the next column, what is it called ? What do you set
down when you come to the last column ?

hundreds. Add the column of hundreds and write down the sum,
and the entire sum is 1535.

~ 2. When the sum, in any column, exceeds 9, it produces one or
more units of a higher order, which belongs to the next column at
the left. In that case, write down the excess over exact tens, and
add to the next column as many units of its own order, as there
were tens in the sum.

This is called carrying to the next column. The number to
be carried, should not, in practice, be written under the col-
umn at the left, but added mentally.

Hence, to find the sum of two or more numbers, we have
the following

RULE.

I. Write the numbers to be added, so that units of the same
order shall stand in the same column.

II. Add the column of units. Set down the units of the
sum and carry the tens to the next column.

III. Add the column of tens. Set down the tens of the sum
and carry the hundreds to the next column ; and so on, till
all the columns are added, and set down the entire sum of the
last column.

PROOF.

34* The proof of any operation, in Addition, consists In
showing that the result or answer contains as many units as
there are in all the numbers added, and no more. There are
two methods of proof, for beginners.*

I. Begin at the top of the units column and add all the
columns downwards, carrying from one column to the other,
as when the columns were added upwards. If the two
results agree the work is supposed to be right. For, it is
not likely that the same mistake will have been made in both

II. Draw a line under the upper number. Add the lower
numbers together, and then add their sum to the upper number.

* NOTE. If the teacher prefers the method of proof by casting
out the 9's, that method, for the four ground rules, will be found
in the University Arithmetic.

84. What does the proof consist of in addition? How many
methods of proof are there? Give the two methods.

SIMPLE NUMBERS.

If the last sum is the same as the svm total, first found, the
work may be regarded as right.

EXAMPLES.

1. What is the sum of the numbers 375,
6321, and 598?

The small figure placed under the 4, shows how
many are to be carried from the units' column, and
the small figure under the 9, how many are to be
carried from the tens' column.

Also, in the examples below, the small figure un-

OPERATION.

375

6321

598

7294
11

der each column shows how many are to be carried to the next
column at the left. Beginners should set down the numbers to be
carried, as in the examples.

Ans. 110012

2221

Ans.

(3.)

9841672
793159

888923

11523754

221111

(4.)
81325
6784
2130

Ans. 90239
1110

(5.)
4096
3271
4722

(6.)
9976

8757
8168

9875
9988

8774

(8.)
67954
98765
37214

(9.)
6412
1091
6741

9028

(10.)
90467
10418
91467
41290

(11.)
87032
64108
74981
21360

(12.)
432046
210491

809765
542137

(13.)

21467

80491

67421

4304

2191

(14.)

89479

75416

7647

214

19

(15.)

74167

21094

2947

674

85

(16.)

9947621

704126

81267

9241

495

28

(17.)

34578

~3750

87

328

17

327

Sum 39087
~4509

Proof 39087

(20.)

672981043

67126459

39412767

7891234

109126

84172

72120

(18.)

22345

67890

8752

340

350

78

Sum 99755

77410
Proof 1)9755

(21.)

91278976

7654301

876120

723456

31309

4871

978

(19.)

23456

78901

23456

78901

23456

78901

Sum 307071

Proof 307071

(22.)

8416785413

6915123460

31810213

7367985

654321

37853

2685

The pupil should be early taught to omit the intermediate wordi
in the addition of columns of figures. Thus, in example 22,
instead of saying 5 and 8 are eight and 1 are nine, he should say
eight, nine, fourteen, seventeen, twenty. Then, in the column of
tens, ten, fifteen, seventeen, twenty-five, twenty-six, thirty-two,
thirty-three. This is called reading the columns. Let the
pupils be often practised in it, both separately, and in concert in
classes.

23. Add 8635, 2194, 7421, 5063, 2196, and 1245 to-
gether.

24. Add 246034, 29S765, 47321, 58653, 64218, 5376,
9821, and 340 together.

25. Add 27104, 32547, 10758, 6256, 704321, 730491,
2587316, and 2749104 together.

26. Add 1, 37, 39504, 6890312, 18757421, and 265 to-
gether.

27. What is the sum of the following numbers, via:
seventy-five; one thousand and ninety-five; six thousand
four hundred and thirty-five; two hundred and sixty-seven

SIMPLE NUMBERS. 29

thousand ; one thousand four hundred and fifty-five ; twenty-
seven millions and eighteen ; two hundred and seventy mil-
lions and twenty-seven thousand ?

28. What is the sum of 372856, 404932, 2704793,
9078961, 304165, 207708, 41274, 375, 271, 34, and 6?

29. What is the sum of 4073678, 4084162, 3714567,
27413121, 27049, 87419, 27413, 604, 37, and 9 ?

30. What is the sum of 36704321, 2947603, 999987, 76,
47213694, 21612090, 8746, 31210496, and 3021 ?

31. Add together fifty-eight billions, nine hundred and
eighty-two mill ions, four hundred and eighty-seven thousands,
six hundred and fifty-four ,- seven hundred and forty billions,
three hundred and fifty millions, five hundred and forty
thousands, seven hundred and sixty ; four hundred and
twenty-five billions, seven hundred and three millions, four
hundred and two thousands, six hundred and three ; thirty-
four billions, twenty millions, forty thousands and twenty ;
five hundred and sixty billions, eight hundred millions, seven
hundred thousands and five hundred.

(32.) (33.) (34.)

87406 92674 25043

89507 27049 97069

41299 28372 81216

47208 37041 75850

71615 49741 90417

72428 57214 19216

97206 59261 20428

41278 41219 60594

28907 57267 72859

325412 3 40216 43706

S 27049 g 87614 g 21441

28416 92742 87604

72204 87046 71215

70412 90212 . 18972

27426 17618 27042

62081 40261 59876

81697 57274 54301

87489 21859 87415

21642 42673 32018

24672 51814 7268T

APPLICATIONS.

35* In all the applications of arithmetic, the numbers ad-
ded together must Imve the same unit.

In the question, How many head of live stock in a field,
there being 6 cows, 2 oxen, 3 steers, and 15 sheep, the unit
is 1 head of live stock. And the same principle is applicable
to all similar questions.

QUESTIONS FOR PRACTICE.

1. HOTT many days are there in the twelve calendar
months? January has 31, February 28, March 31, April
30, May 31, June 30, July 31, August 31, September 30,
October 31, November 30, and December 31.

Ans.

2. What is the total weight of seven casks of merchandise ;
No. 1, weighing 960 pounds, No. 2, 725 pounds, No. 3,
830 pounds, No. 4, 798 pounds, No. 5, 698 pounds, No. 6,
569 pounds, No. 7, 987 pounds ?

3. At the Custom House, on the 1st day of June, there
ir ere entered 1800 yards of linen; on the 10th, 2500 yards;
on the 25th, 600 yards; on the day following, 7500 yards;
and the last three days of the month, 1325 yards each day :
what was the whole amount entered during the month ?

Ans.

4. A farmer has his live-stock distributed in the following
manner: in pasture No. 1, there are 5 horses, 14 cows, 8
oxen, and 6 colts ; in pasture No. 2, 3 horses, 4 colts, 6 cows,
20 calves, and 12 head of young cattle; in pasture No. 3,
320 sheep, 16 calves, two colts, and 5 head of young cattle.
How much live-stock had he of each kind, and how many

Ans. horses, cows, oxen, colts, calves,
head of young cattle, and sheep.