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CONTENTS.

NOTATION by Roman Letters. . Page 51 Loss and Gain...........

98

Notation and Addition.....

6 Equation of Payments

102

Addition

7 Barter......

103

Subtraction

13 Custom House Allowances.. 105

Multiplication................. 17 Method of Assessing Taxes.. 107

Division...

23 Vulgar Fractions..

Problems and Miscellaneous Quest Reduction of Vulgar Fractions.... 111

tions, involving the principles of Addition of Vulgar Fractions..... 117

the preceding Rules....

32 Subtraction of Vulgar Fractions .. 118

Federal Money...

34 Multiplication of Vulgar Fractions 119

Reduction of Federal Money.. ib. Division of Vulgar Fractions ... 120

Addition of Federal Money. 35 Single Rule of Three in Vulgar

Subtraction of do. do.

36 Fractions

121

Multiplication of do. do.

ib. Double Rule of Three in Vulgar

Division of do. do.

37 Fractions..

123

Table of Coins of the U. S. and Involution

....... 124

their Reduction....

38 Evolution.

Reduction ....

39 Extraction of the Square Root.... 126

Tables of Weight, Measure, &c. 39 to 48 Do. of the Cube Root...... 127

Reduction of Gold Coins.

ib. Do. of the Roots of all Pow.

Decimal Fractions....

49

ers.

130

Compound Addition

50 Further Use of the Square Root... 133

Do. Subtraction.

53 Further Use of the Cube Root..... 134

Do. Multiplication

58 Table of Foreign Coins...........

136

Do. Division ......

60 Exchange

ib.

Explanation, Notation, and Nume. Exchange with Great Britain 137

ration of Vulgar Fractions...... 61 Bills of Exchange, both above and

Reduction of Vulgar Fractions to

below par.::

138

Decimal...

ib. Exchange with France. .

139

Reduction of Compound Numbers Do. with Spain

140

to a Decimal ....

62

with Portugal

142

To find the value of a Decimal, &c. ib.

Do. with Holland, Hamburg,

Multiplication and Practice. 63 Russia, and China

144

To find the value of articles sold by Supplement to Cubic Measure..... 145

the 100 or 1000......

65 Cubic and Square Measure.... 146

To find the largest common divisor, Multiplication Contracted. And
or measure .........

ib. Difference of Longitude given,

To find the least common multiple to find the Difference of Time... 147

of two or more numbers.... 66 To find the Area of a globe or ball,

Ratio and Proportion

67 casks, &c....

148

Rule of Three, or Single Proportion ib. Alligation..

149

The Double Rule of Three, or Com Permutation and Combination.... 151

pound Proportion

173 Arithmetical Progression

153

Interest

75 Geometrical Progression

154

Discount

83 United States' Duties

155

Compound Interest

85 Single Position

157

Cubic Measure..

86 Double Position

159

Squarę do.

87 Tonnage of Ships, &c.

160

Oblong Square.

89 Gauging..

162

Paving and Plastering

ib. Annuities at Compound Interest .. 165

Shingle, or Roof Measure.

90 Annuities, Leases, &c. taken in re-

Circle Measure ...

91 version at Compound Interest... 169

Round Timber, &c.

93 Perpetuities at Compound Interest 170

Measurement of Stone in a Well. 94 Perpetuities in Reversion........ 171

Fellowship

95 Miscellaneous Examples..........

172

Do.

ARITHMETIC is the art and science of computing by numbers: the rules upon which all its operations depend, are Notation, Numeration, Addition, Subtraction, Multiplication, and Division.

NOTATION. NOTATION teaches to write and express words by the ten Arabic characters, called figures, or digits : viz.

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are

NOTATION BY LETTERS. To learn the following table, begin at the left hand column, and read thus: one I. is one, two II. are two, three III. are 3, IV. are 4, &c. One I. is 1. XX.

20. 11. 2. XXX.

30. III.

3.
XL.

40.
IV.
4. L.

50. V. 5. LX.

60. VI. 6. LXX.

70. VII. 7. LXXX.

80. VIII. 8. XC.

90. IX. 9.

100. X. 10. CC.

200. XI. 11. CCC,

300. XII. 12. CCCC.

400. XIII. 13. D.

500. XIV. 14. DC.

600. XV. 15. DCC.

700. XVI. 16. DCCC.

800. XVII. 17. DCCCC.

900. XVIII. 18. M.

1000. XIX.

19.

MDCCCXXIX. 1829.

C.

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NOTATION. By the ten arabic figures all numbers are expressible. The figure in the first place, reckoning from right to left, denotes only its simple value; that in the second place, ten times its simple value; that in the third, a hundred times its simple value, and so on; always ten times its former value.

Thus, write the figures five thousand eight hundred and thirtyfour (5834); the 4 in the first place counts four; the second figure

3-counts thirty; 8 in the third place eight hundred; and 5 in the fourth place five thousand. A cipher counts nothing by itself; but annexed (to a whole number) increases its value in a tenfold proportion: thus 6 counts only six; but place a cipher to the right hand, thus 60, and it reads sixty.

EXAMPLES. Write in figures the following numbers: Sixteen. Twenty-two. Forty-four. Seventy-five. One hundred and twenty. Six hundred and two. One thousand one hundred and eleven. Sixteen thousand seven hundred and seven. Two hundred twelve thousand and eight hundred. One million one hundred eleven thou

sand one hundred and ten. Ten million ten hundred thousand and ten hundred. Eight hundred seventy-six millions, five hundred forty-three thousand and ten; &c. &c.

Questions.-What is arithmetic? What is an art? (An art is a collection of rules and precepts for doing a thing with ease and accuracy; an art is knowledge in practice, as weaving or gardening.) What is science ? (Science is a system of any branch of knowledge, comprehending its doctrine, reason, and

theory; it is knowledge in theory, as theology, or physic.) What
is notation? What are the names of the ten Arabic figures ?

NUMERATION.
NUMERATION teaches the reading of any number (or series) of
figures.
NUMERATION TABLE, SHOWING THE PLACE OF

1 units,
21 tens,
321 hundreds,
4,321 thousands
54,321 tens of thousands,
654,321 hundreds of thousands,
7,654,321 millions,
87,654,321 tens of millions,
987,654,321 hundreds of millions,
9,987,654,321 thousands of millions,
99,987,654,321 tens of thousands of millions,
999,987,654,321 hundreds of thousands of millions,
9,999,987,654,321 billions,
99,999,987,654,321 tens of billions,
999,999,987,654,321 hundreds of billions,
9,999,999,987,654,321 thousands of billions,
99,999,999,987,654,321 tens of thousands of billions,

999,999,999,987,654,321 hundreds of thousands of billions.
Octillions Septillions Sectillions Quintillions Quadrillions Trillions
222,222 222,222 222,222 222,222

222,222 222.222

Questions. What is Numeration ?

How must figures be numerated ? In what manner should figures be read? Why do we numerate figures from the right hand to the left? In what proportion do they increase in value? How many units are there in ten? How many tens in a hundred? How many in a thousand ? · How many hundred in a thousand? How many thousand in a million ?

ADDITION. The following numbers may be put to the pupil in separate questions, by the teacher, thus, How many are 1 and 1.

1 and 1 2 and 13 and 14 and 15 and 16 and 1 2 1 2 23 24 25 26 2 3 1 3 213 314 35 36 3 1

23 414 4 5 416 4 1 2 3 54 5

5 16 5 1 6 23 64 65 6 16 7 1 7 2) 3 714 75 7

7 8 1 8 23 84 85 8 6 8 9 1 9 23 94 95 96 9 10 1 | 10 23 10.14 10 1 5 10 16 10 11 111 23 114 11 / 5 116 11 12 112 23 124 125 126 12

&c. &c. &c. &c. &c. &c.

1. If you have two apples in one hand, and one in the other, and four in your pocket, how many have you in all ?

2. James gave 8 cents for a purse and had 6 cents left to put in it: how many cents had he at first?

3. How many are 12 and 8?

4. William paid 8 cents for a copy-book, 10 cents for an inkstand, and 3 cents for quills: how many cents did he pay out?

5. Charles bought a hat for 2 dollars, a coat for 7 dollars, and pantaloons for 4 dollars; how many dollars did the three cost him?

6. Samuel paid 3 cents for candy, 4 cents for apples, and 10 cents for a primer: how many cents did he pay out?

6+9+5= how many?
2+12+8= how many ?
14+6+8+2= how many?
20+10+0+5= how many?
100+5+10+7= how many?
30+0+1+9+6= how many?

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ADDITION. ADDITION is the adding of two or more numbers into one sum total, or amount.

RULE. Set the given numbers under each other, with units under units, tens under tens, hundreds under hundreds, &c. Then draw a line under the lowest number, and begin at the units or right hand column, add all the column together-set down the sum when less than ten; if ten, or more, set down the right hand figure, and add the left (hand figure) to the next column: and thus proceed to the last column, and set down the whole amount of it.

PROOF Perform the operation a second time, agreeably to the rule ; but in one case begin at the bottom, and in the other at the top. Or,

Reserve one of the given numbers, find the sum of the rest, and thereto add the number reserved.

NOTE.-The reason of carrying one for every ten is evident from what has been taught in Notation, because ten in any column is just equal to one in the next left hand column.

ADDITION TABLE. Read it thus: 2 and 2 are 4: 2 and 3 are 5, &c. 2+2=4 3+8=11 5+7=12 7+11=18 3 5 9 12 8 13

12 19 4 6 10 13

9 14

8+8=16 57 11 14 10 - 15

9 17
6 8 12 15 11 16

10 18.
7 9
4+4= 8 12

17

11 19
8 10

5 9 6+6=12 12 20
9 11
6 10

7 13 9+9=18
10 12

7 11 8 14 10 19
11 13
8 12
9 15

11 20
12 14
9 13 10 16

12 21
3+3=6 10 14 11 17 10+10=20
4 7 11 15 12 18

11 21 5 8

16 7+7=14 12 22
6 9 5+5=10

8 "15 11+11=22
7 10
6 11

9 16 12 23
10 17 12+12=24

12

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