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Numeration, 9|General Method of Making Taxes, 140

— Table, 10|Rule of Three Direct, 142

Notation by Roman Letters, 12. Inverse, 143

Addition, 13|Double Rule of Three, 143

— Table, 14|Conjoined Proportion, 146

Subtraction, 17|Practice, 148

Multiplication, 20|Single Fellowship, - 154

—— Table, 21|Double Fellowship, 156

Division, - 27|Custom-House Allowances, 157

—- Table, 28American Duties, 160

Federal Money, 33Discount, 161

Table of American Coins, 33|Barter, 163

Do. of Foreign Coins, 39|Loss and Gain, 165

Compound Addition, 40|Equation of Payments, 166

—— Subtraction, 52 Involution, 168

Problems, resulting from a comparison Evolution, 168

of the preceding rules, 56|Table of Powers, 169

Reduction, 59|Extraction of the Square Root, 170

Vulgar Fractions, 70|Extraction of the Cube Root, 178

Decimal Fractions, 89 Rule for extracting the roots of all

Compound Multiplication, 99. powers, 184

Division, 105|Alligation Medial, 185

Simple Interest, 311||Alligation Alternate, 186

Commission, 117|Single Position, 191

Buying or Selling Stocks, 118' Double Position, 192

Insurance, 119. Permutation, 194

Simple Interest by Decimals, #. 195

Compound Interest, 121|Mechanical Powers—Of the Lever, 197

- — by Decimals, 123 Of the Wheel and Axle, 197

Duodecimals, or Cross Multiplication, 124 Of the Screw, 198

The Single Rule of Three, 130|Useful and Diverting Questions, 199

in PEx To THE TABLES.

Addition, looses: by decimals, 119, 123

Ale or Beer Measure, 50||and or Square Measure, 48

Aliquot parts of money weight, Long do. 45, 46

C. 112, 143, 149 Money, 33, 41

American Coins, 33|Motion, 47
Apothecaries' Weight, #Mulliplication, 21
Avoirdupois do. 43|Notation, 12
Cloth Measure, 44||Numeration, 10
Decimals, 89 Pence and shillings, 41
Decimal parts of a year, 120 Powers. 169
Discount per cent. 149 Solid Measure, 49
Division, 28 Square do. 48
Dry Measure, 51|Taxes, for making, 141
Duodecimals, - 124 Time, o 46
English Money, 41|Troy Weight, 42
Federal Money, 33 Wine Measure, 49

Foreign Coins, 39

[[P Beside the tables above enumerated, the scholar will find in the CIPHERING-
Book, published in connexion with this volume, a variety of ve:y usesul tables, and
several not found in common arithmeticks.

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Signs.

"Two parallel horizontal lines are the sign of equality. It shows that the number before, is equal in value to the number after it. Example, 1 dollar=100 cents, is read thus, 1 dollar is equal to 100 cents.

+ Two short lines, crossing each other at right angles, are the sign of Addition. It shows that numbers with this sign between them, are to be added together. Erample, 5+7=12, is read thus, 5 added to 7, or 5 plus 7, is equal to 12.

— A short horizontal line is a sign of Subtraction. It shows that the number after it, is to be taken from the number before it. Example, 12–7–5, is read thus, 12 less 7, or 12 minus 7, is equal to 5.

× Two short lines crossing each other in the form of an X, are the sign of Multiplication. It shows that the number before it, is to be multiplied by the number after it. Example, 6×5=80, is read thus, 6 multiplied by 5, is equal to 30.

+ A short horizontal line between two points, is the sign of Division. It shows that the number before it, is to be divided by the number after it. Example, 30+6 =5, is read thus, 80 divided by 6, is equal to 5.

:: Four double points or colons are the sign of Proportion ; and to show that numbers are proportional, they are written thus, 2 : 4 :: 8 : 16, which are read, 2 is to 4 as 8 is to 16.

To This sign signifies the second power or square.

To This sign signifies the third power or cube. 2

A/ This sign, prefixed to any number, shows that the square root of the number is required.

V This sign, prefixed to any number, shows that the cube root of the number is required.

A R IT H M ETIC K.

ARITHMErick is the science of Numbers;* and comprises the following principal rules, viz. I. Notation, or NUMERATION ; II. Addition ; III. SUBTRAction ; IV. MULTIPLicATION ; and W. Division. The four last are called simple, when the numbers are all of one denomination ; compound, when the numbers are of different denominations.

These five rules are called principal or fundamental, because the whole art of arithmetick is comprehended in their various operations.

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1. NUMERATION teaches us to read or write any sum or number, by means of the following ten characters, called figures :f

Cipher. One. Two. Three. Four. Five. Six. Seven. Eight. Nine,

0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

, 2. The value of a figure, when alone, is called its simple value, and is invariable. Figures have also a local value, which varies

* Number is either a unit, or a collection of units. A single thing is a unit, or one. One and one are two. One and two are three. And thus, by constant addition, all numbers are generated.

# These figures, which are of Arabic or Indian origin, were introduced into Europe by the Moors, about the year 1150; they were formerly all called ciphers, whence it came to pass that the art of arithmetick was called ciphering. -

1. What is Arithmetick 2–2. What are called its fundamental rules 3–3. What does numeration teach 7–4. How many figures are used to represent numbers ?— 5. How do you determine their value?

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according to the place in which they stand. In a combination of figures, reckoning from the right hand to the left, the figure at the right, or in the first place, represents its simple value ; that in the second place, ten times the simple value ; that in the third place, ten times the value of that in the second place ; and so

on, in a tenfold increase.

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The first figure at the

right, in the place of units, has its simple value, or the same as if standing alone—four. The second, in the place of tens, signifies four tens, or forty. The third figure, in the place of hundreds, signifies four hundred, or ten times its value in the place of tens. The fourth figure is in the place of thousands, bearing ten times its value in the

place immediately preceding.

3. A cipher 0 though of no signification itself, when placed on the right hand of figures, in whole numbers, increases their value in the same tenfold proportion. Thus 9, signifies only nine, its simple value. Place a cipher on the right, (90) it becomes ninety ; and by placing two ciphers at the right, thus, (900) it be

comes nine hundred.

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Note.—Six places of figures, beginning on the rfght, are called a period ; but they are commonly divided into half periods of three

6. In a combination of figures, how do you ascertain their value?—7. What is the nature of the cipher 7–8. What is the period of JNumeration ?

figures each. This division enables us to read any number of figures as easily as we can read the first period. *

RULE.—Commit the words at the head of the Table, viz. units, tens, hundreds, &c. to memory; then, to the simple value of each figure, join the name of its place, beginning at the left hand and reading towards the right. More particularly—1. Place a dot under the right hand figure of the 2d, 4th, 6th, 8th, &c. half periods, and the figure over such dot will, universally, have the name of thousands2. Place the figures 1, 2, 3, 4, &c. as indices, over the 2d, 3d, 4th, &c. period: These indices will then show the number of times the millions are involved—the figure under 1 bearing the name of millions, that under 2, the name of billions, or millions of millions;–

that under 3, trillions, or millions of millions of millions.

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- - - - • * *

te.—Billions is substituted for millions of millions ; Trillions, for millions of millions of millions; Quatrillions, for millions of millions of millions of millions. Quintillions, Sextillions, Septillions, Octillions, Nonillions, Decillions, Undecillions, Duodecillions, &c. answer to millions so often involved as their indices respectively denote.

The right hand figure of each halfperiod has the place of units, of that half period; the middle one, that of tens, and the left hand one that of hundreds.

APPLICATION.—Let the scholar now read, or write down in words at length, the following numbers :- 8 437 709,040 3.476.194 7.184,397.647 17 3.010 879.096 84,094.007 49. 163.189, 186 129 76.506 4.091.875 690.748.591 500.098,422.700.

Write down, in proper figures, the following numbers — Fifteen - - - - - -

Two hundred and seventy-nine -

9. How are numbers commonly divided ?––10. Of what use is this division ?—11, What is the rule for numeration ?

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