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| 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............ ... ... 110 !
Problems and Miscellaneous Ques- Reduction of Vulgar Fractions.... 111

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

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Subtraction of Vulgar Fractions ... 118

|Multiplication of Vulgar Fractions 119

Division of Vulgar Fractions ..... 120

Single Rule of Three in Vulgar

Fractions . . . . . . . . . . . . . . . . . . . . . . 121

Double Rule of Three in Vulgar

Fractions . . . . . . . . . . . . . . . . . . . . . .

Involution . . . . . . . . . . . . . . . . . . . . . . . 124

Evolution .......... . . . . . . . . . . . . . . 125

Extraction of the Square Root.... 126

Do. of the Cube Root...... 127

Do. of the Roots of all Pow-

€rs . . . . . . . . . . . . . . . . - - - - - - - - - - - - 130

Further Use of the Square Root... 133

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

Table of Foreign Coins........... 136

Exchange . . . . . . . . . . . . . . . . . . . . . . . . ib.

Exchange with Great Britain . . . . . 137

Bills of Exchange, both above and

below par.... . - - - - - - - - - . . . . . . . . . 138

Exchange with France. . . . . . . . . . . 139

Do. with Spain . . . . . . . . . . . . . 140

Do. with Portugal .......... 142

Do., with Holland, Hamburg,

Russia, and China.......... .... 144

Supplement to Cubic Measure..... 145

Cubic and Square Measure........ 146

Multiplication Contracted. And

Difference of Longitude given,

to find the Difference of Time... 147

To find the Area of a globe or ball,

casks, &c. . . . . . . . . . . . . .......... 148

Alligation. . . . . . . . . . . . . . . - - - - - - - - - 149

Permutation and Combination. ... 151

Arithmetical Progression . . . . . . . . . 153

Geometrical Progression .......... 154

United States' Duties. ....... . . . . . 155

Single Position ........ . . . . . . . . . . . 157

Double Position .................. 159

Tonnage of Ships, &c. . . . . . . . . . . . . 160

Gauging. . . . . . . . . . . . . . . . . . . . . . . . . . 162

Annuities at Compound Interest ... 165

Annuities, Leases, &c. taken in re-

version at Compound Interest... 169

Perpetuities at Compound Interest 170

Perpetuities in Reversion..... .... 171

Miscellaneous Examples. . . . . . . . . . 172

ARITHMETIC.

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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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. are 20. - II. 2. - XXX. 30. III. 3. XL. 40.

IV. 4. L. 50.

W. 5. LX. 60.

WI. 6. LXX. - 70.

VII. 7. LXXX. 80.

VIII. 8. XC. 90.

IX. 9. C. 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.

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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

en 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

21 tens, -
321 hundreds,
4,321 thousands
54,321 tens of thousands,
ar 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,
999.999999.987,654,321 hundreds of thousands of billions.

ons Octillions Septillions Sectillions Quintillions Quadrillions Trillions 222,222 222,222 222,222 222,222 222,222 222,222

sand one hundred and ten. Ten million ten hundred thousand and #

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8 ADDITION.

ADDITION.

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

Tru LE.

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.

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5 7 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 || 6-1-6=12 12 20

• 9 11 6 10 7 13 9-H 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 | 10-1-10=20 4 7 11 15 12 18 11 21

5 8 12 16 || 74-7–14 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

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