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Reduction of Currencies,

To reduce English, &c. Currencies to Federal Money,
Federal Money to the Currencies of England, &c.
one Currency to the Par of another Currency,

Interest,
Time, Rate per cent., and Amount given, to find the Principal,
Time, Rate per cent., and Interest given, to find the Principal,
Principal, Interest, and Time given, to find the Rate per cent.,
Principal, Rate per cent., and Interest given, to find the Time,

To find the Interest on Notes, Bonds, &c., when partial Payments have

been made,
Compound Interest,

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151

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153

154

155

156

164

165

166

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by Progression,

229

Equation of Payments,

176

Ratio, or the Relation of Numbers,

177

Proportion, or Single Rule of Three,

179

Same Questions, solved by Analysis, ¶ 65, ex. -20.

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Scale for taking Dimensions in Feet and Decimals of a Foot, 204

Involution,

205 | Evolution,

207

Extraction of the Square Root,

207

Application and Use of the Square Root, see Supplement,

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To find the Area of a Square or Parallelogram, ex. 148–154.

of a Triangle, ex. 155-159.

Having the Diameter of a Circle, to find the Circumference; or, having the

Circumference, to find the Diameter, ex. 171-175.

To find the Area of a Circle, ex. 176-179.

of a Globe, ex. 180, 181.

To find the Solid Contents of a Globe, ex. 182-184.
of a Cylinder, ex. 185-187.

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ARITHMETIC.

NUMERATION.

1. A SINGLE or individual thing is called a unit, unity or one; one and one more are called two; two and one more are called threc; three and one more are called four; four and one more are called five; five and one more are called six; six and one more are called seven; seven and one more are called eight; eight and one more are called nine; nine and one more are called ten, &c.

These terms, which are expressions for quantities, are called numbers. There are two methods of expressing numbers shorter than writing them out in words; one called the Roman method by letters, and the other the Arabic method by figures. The latter is that in general use.

*

In the Arabic method, the nine first numbers have each an appropriate character to represent them. Thus,

*In the Roman method by letters, I represents one; V, five; X, ten; L, fifty: C, one hundred; D, five hundred; and M, one thousand.

As often as any letter is repeated, so many times its value is repeated, unless it be a letter representing a less number placed before one representing a greater then the less number is taken from the greater; thus, IV represents four, IX, ning &c., as will be seen in the following

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*I is used instead of D to represent five hundred, and for every additional nexed at the right hand the number is increased ten times.

an

CIO is used to represent one thousand, and for every C and Ɔ put at each end, the

dumber is increased ten times.

A line over any number increases its value one thousand times

A unit, unity, or one, is represented by this character,

Two

Three

Four

Five

Six

Seven

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Eight
Nine

Ten has no appropriate character to represent it; but is
considered as forming a unit of a second or higher
order, consisting of tens, represented by the same
character (1) as a unit of the first or lower order,
but is written in the second place from the right
hand, that is, on the left hand side of units; and
as, in this case, there are no units to be written
with it, we write, in the place of units, a cipher, (0,)
which of itself signifies nothing; thus,
One ten and one unit are called

Ten

10

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Eleven

11,

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Ten tens are called a hundred, which forms a unit of a still higher order, consisting of hundreds, represented by the same character (1) as a unit of each of the foregoing orders, but is written one place further toward the left hand, that is, on the left hand side of tens; thus, One hundred 100. One hundred, one en, and one unit, are called

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One hundred and eleven 111

T2. There are three hundred sixty-five days in a year. In this number are contained all the orders now described, viz. units, tens, and hundreds. Let it be recollected, units Occupy the first place on the right hand; tens, the second place from the right hand; hundreds, the third place. This number may now be decomposed, that is, separated into parts, exhibiting each order by itself, as follows:-The highest order, or hundreds, are three, represented by this character, 3; but, that it may be made to occupy the third place, counting from the right hand, it must be followed by two ciphers, thus, 300, (three hundred.) The next lower order, or tens, are six, (six tens are sixty,) represented by this character, 6; but, that it may occupy the second place, which is the place of tens, it must be followed by one cipher, thus, 60, (sixty.) The lowest order, or units, are five, represented by a single character, thus, 5, (five.)

We may now combine all these parts together, first writing down the five units for the right hand figure, thus, 5; then the six tens (60) on the left hand of the units, thus, 65; then the three hundreds (300) on the left hand of the six tens, thus, 365, which number, so written, may be read three hundred, six tens, and five units; or, as is more usual, three hundred and sixty-five.

3. Hence it appears, that figures have a different value according to the PLACE they occupy, counting from the right hand towards the left.

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Take for example the number 3 3 3, made by the same figure three times repeated. The 3 on the right hand, or in the first place, signifies 3 units; the same figure, in the second place, signifies 3 tens, or thirty; its value is now increased ten times. Again, the same figure, in the third place, signifies neither 3 units, nor 3 tens, but 3 hundreds, which is ten times the value of the same figure in the place immediately preceding, that is, in the place of tens; and this is a fundamental law in notation, that a removal of one place towards the left increases the value of a figure TEN TIMES.

Ten hundred make a thousand, or a unit of the fourth order. Then follow tens and hundreds of thousands, in the same manner as tens and hundreds of units. To thousands

succeed millions, billions, &c., to each of which, as to units and to thousands, are appropriated three places,* as exhibited in the following examples:

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EXAMPLE 2d.

3 17 4 5

3 7 4 6 3 5

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

To facilitate the reading of large numbers, it is frequently practised to point them off into periods of three figures each, as in the 2d example. The names and the order of the periods being known, this division enables us to read numbers consisting of many figures as easily as we can read three figures only. Thus, the above examples are read 3 (three) Quadrillions, 174 (one hundred seventy-four) Trillions, 592 (five hundred ninety-two) Billions, 837 (eight hundred thirty-seven) Millions, 463 (four hundred sixtythree) Thousands, 512 (five hundred and twelve.)

After the same manner are read the numbers contained in the following

*This is according to the French method of counting. The English, after hundreds of millions, instead of proceeding to billions, reckon thousands, tens and hundreds of thousands of millions, appropriating six places, instead of three, to illions, billions, &c

Units.

of Units.

e Units

o Hundreds - Tens

~ Units

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