application of these to the succeeding rules; and, besides, will serve to interest him in the science, since he will find himself able, by the application of a very few principles, to solve many curious questions.

The arrangement of the subjects is that, which to the author has appeared most natural, and may be seen by the Index. Fractions have received all that consideration which their importance demands. The principles of a rule called Practice are exhibited, but its detail of cases is omitted, as unnecessary since the adoption and general use of federal money. The Rule of Three, or Proportion, is retained and the solution of questions involving the principles of proportion, by analysis, is distinctly shown.

The articles Alligation, Arithmetical and Geometrical Progression, Annuities and Permutation, were prepared by Mr. IRA YOUNG, a member of Dartmouth College, from whose knowledge of the subject, and experience in teaching, I have derived important aid in other parts of the work.

The numerical paragraphs are chiefly for the purpose of reference: these references the pupil should not be allowed to neglect. His attention also ought to be particularly directed, by his instructer, to the illustration of each particular principle, from which general rules are deduced for this purpose recitations by classes ought to be instituted in every school where arithmetic is taught.

:

The supplements to the rules, and the geometrical demonstrations of the extraction of the square and cube roots, are the only traits of the old work preserved in the new.

Mont Vernon, (N. H.) Sept. 29, 1827.

DANIEL ADAMS.

Miscellaneous Questions, involving the Principles of the preceding Rules,

Different Denominations,

Federal Money,

Reduction,

COMPOUND NUMBERS.

to find the Value of Articles sold by the 100, or 1000,
Bills of Goods sold,

Tables of Money, Weight, Measure, &c.

Addition of Compound Numbers,

Subtraction,

Multiplication and Division,

FRACTIONS.

COMMON, OF VULGAR. Their Notation,

Proper, Improper, &c.

To change an Improper Fraction to a Whole or Mixed Number, a Mixed Number to an Improper Fraction,

To reduce a Fraction to its lowest Terms,

Greatest common Divisor, how found,

To divide a Fraction by a Whole Number; two ways, To multiply a Fraction by a Whole Number; two ways, a Whole Number by a Fraction,

one Fraction by another,

General Rule for the Multiplication of Fractions,
To divide a Whole Number by a Fraction,
one Fraction by another,

Multiplication of Decimal Fractions,

To reduce Vulgar to Decimal Fractions,

Reduction of Decimal Fractions,

145

To reduce Shillings, &c., to the Decimal of a Pound, by Inspection,

146

the three first Decimals of a Pound to Shillings, &c., by Inspection, 157

General Rule for the Division of Fractions,
Addition and Subtraction of Fractions,

Common Denominator, how found,

Least Common Multiple, how found,

Rule for the Addition and Subtraction of Fractions,

Reduction of Fractions,.

DECIMAL. Their Notation,

Addition and Subtraction of Decimal Fractions,

Division of Decimal Fractions,

Page

Equation of Payments,

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,

by Progression,

Ratio, or the Relation of Numbers,

Proportion, or Single Rule of Three,

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

Compound Proportion, or Double Rule of Three,

Fellowship,

'T'axes, Method of assessing,

151

153

154

155

156

164

165

166

167

168

169

229

176

177

179

187

192

195

Alligation,

197

Duodecimals,

201

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

Application and Use of the Square Root, see Supplement,

Application and Use of the

Arithmetical Progression,
Annuities at Compound Interest,
Practice, 29, ex. 10-19. T 43.
Insurance, 182.

Buying and Selling Stocks, ¶ 82.

Cube Root, see Supplement,
222 Geometrical Progression,
231 Permutation,

Commission, 82; ¶ 85, ex. 5, 6.
Loss and Gain, ¶ 82; ¶ 88, cx. 1—8.
Discount, ¶ 85, ex. 6-11.

212

215

220

225
237

MISCELLANEOUS EXAMPLES.

Barter, ex. 21-32.

| Position, ex. 89-108.

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, cx. 182-184.
of a Cylinder, ex. 185–187.

ARITHMETIC.

NUMERATION.

11. 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 three; 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, nine, &c., as will be seen in the following

* 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 number 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

1.

2.

3.

4.

5.

6.

7.

8.

9.

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
One ten and two units are called
One ten and three units are called
One ten and four units are called
One ten and five units are called
One ten and six units are called
One ten and seven units are called
One ten and eight units are called
One ten and nine units are called
Two tens are called
Three tens are called

Four tens are called
Five tens are called
Six tens are called
Seven tens are called

Eight tens are called

Nine tens are called

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 ten, and one unit, are called One hundred and eleven 111.