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Why the given number is separated into periods-Principle on which

the first figure in the root is doubled for a divisor in finding the next figure in

the root (299).

How integers and decimals are separated into periods, in extracting the square

root (301)-Number of decimal figures in the root-In finding the root of an im-
perfect square, how the operation may be continued-When a Fraction will be
an imperfect square-Its root how found-How the root may be extracted when
a vulgar fraction is annexed to an integer.

Why each decimal period must contain two figures, and the number of de-
cimal figures in the square root be equal to the number of decimal periods.
Square root of an approximate decimal (302)-The side of a square found from
its area (303).

ARITHMETIC.

CHAPTER I.

PRELIMINARY DEFINITIONS.-NUMERATION.-NOTATION.

Science and Art.

§ 1. SCIENCE is knowledge reduced to a system, so as to be conveniently taught, and readily applied.-ART is knowledge applied to practical purposes.-The Rules of art are founded on the Principles of science.

Unity and Numbers.-Quantity.

§ 2. A Unit is any thing regarded simply as one; and Numbers are repetitions of a unit.

The numbers two, three, &c., are repetitions of a unit or one.

§ 3. Quantity is any thing which admits of being measured. Thus a line is a quantity, and we express its measure when we say it is so many feet or inches long.

Is time a quantity? Is industry a quantity ?
Is hope a quantity? Is distance a quantity ?

Is weight a quantity?
Is virtue a quantity ?

Are length, breadth, and height quantities?

Numbers are quantities, since every number is necessarily measured by a unit, (§ 2); and numbers are necessary to express the measures of all other quantities.

Thus we express the measure of a quantity of water by saying it is five gallons; and the measure or weight of a quantity of iron by saying it is ten pounds.

PRELIMINARY DEFINITIONS.

$ 4. MATHEMATICS is the science of Quantity. Its most general divisions are Arithmetic and Geometry.

ARITHMETIC is the science of Numbers; or, when practically applied, the art of Calculation.-GEOMETRY is the science which treats of Extension, including length, breadth, and height or depth.

Of these two divisions of mathematical science, Arithmetic comes first in order; and we begin by distinguishing different kinds of numbers.

Abstract and Concrete Numbers.

5. An abstract number is a number without any kind of units expressed; as the numbers one, five, ten, a hundred.

§ 6. A concrete number is a number of some kind of units expressed; as the numbers one book, five men, a hundred dollars.

Is twenty an abstract or a concrete number? Is nine pounds an abstract or a concrete number? One hundred? Two hundred miles?

Give two other examples of abstract, and two of concrete numbers.

Similar and Dissimilar Numbers.

$7. Similar concrete numbers are such as express the same kind of units; as three dollars and five dollars.

§ 8. Dissimilar concrete numbers are such as express different kinds of units; as three dollars and five miles.

Are four inches and seven inches, similar or dissimilar concrete numbers? Nine pounds and twelve yards? One cent and ten pints?

Give another example of similar concrete numbers. Of dissimilar concrete numbers. Give another example of each kind.

The groundwork of a thorough knowledge of Arithmetic must be laid in the principles of Numeration and Notation; for on these principles depend the four fundamental operations in Arithmetic-Addition, Subtraction, Multiplication, and Division.

NUMERATION.

§ 9. NUMERATION is the method of naming abstract numbers. Numbers are named as so many units, tens, hundreds, &c. Thus eleven is one and ten;

twelve is two and ten;

thirteen is three and ten;

seventeen is seven and ten.

Twenty is two tens ;
twenty-five is two tens and five;
thirty-nine is three tens and nine.
A hundred is ten tens;

A thousand is ten hundred.

A million is ten hundred thousand, or a thousand thousands. A billion is ten hundred millions, or a thousand millions. A trillion is ten hundred billions, or a thousand billions.

In like manner Numeration proceeds through quadrillions, quintillions, sextillions, septillions, octillions, nonillions, decillions, undecillions, duodecillions, &c.

What two numbers are implied in the name fourteen? In the name fifteen? In eighteen? In fifteen? In seventeen ?

What is implied in the name twenty? In the name twenty-one? In thirty? In forty? In fifty? Sixty? Seventy? Ninety?

A quadrillion is how many? A quintillion? A sextillion? A septillion? An octillion? A nonillion? A duodecillion?

Different Orders of Units.

§ 10. The naming of numbers by tens, hundreds, &c., introduces different orders of units in Numeration.

The numbers two, three, four, &c., are repetitions of the simple unit one, which is called a unit of the first order.

Twenty, thirty, forty, &c., are, respectively, two tens, three tens, four tens, &c.; and in these repetitions of ten, ten is regarded as a unit of the second order.

So in repetitions of a hundred, as two hundred, three hundred, &c., one hundred is regarded as a unit of the third order. In like manner, one thousand is made a unit of the fourth order; and so on.

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