divided by 2 equals 6. An inverted perenthesis also denotes division.

Four points set in the middle of four numbers, denote the numbers to be proportional to one another, in the Rule of Three; as 3: 6:: 12: 24. That is, as 3 are :

A square signifies that the square of any number is required; as, 4-16; or the square of 4 equals 16. This reversed figure seven signifies that the Square Root of a number is required, as, 16-4; that is to say, the Square Root of 16 equals 4. or is 4.

These characters are used to save written words, and to show when to add, subtract, multiply or divide, with out a tedious series of writing. The learners may copy them several times on their slates. See Card No. 26.

DECIMALS.

Having given a hint of decimals in multiplication, I will now insert a few lessons to aid the learner in common business; but as the whole subject of fractions is tedious and but little in use, shall give a view of the most necessary cases only.

What is a fraction?

It is some part of an integer; that is, some part of a whole number; as,

One fourth. One half. Three fourths. Seven eighths.

Nineteen twentieths, &c. 18.

These are called vulgar fractions; and for ease, plainness, and despatch of business, decimals have been invented for a substitute. Therefore our first

business will be to form fractions into decimals.

In order to understand ourselves, let us have a name for each part of the aforegoing fractions.

The upper number is called Numerator; and is made by a remainder after division.

The lower number is called Denominator; and is the divisor in Division. Thus, divide 17 cents equal

ly between 4 boys, each share is 4; that is, four and one fourth.

To turn fractions into decimals, observe the following

RULES.

1. Annex, that is, place on the right side of the Numerator, one or more ciphers, and call it a Dividend.

2. Take the lower number, the Denominator, for a divisor The quotient will be the decimal required. KEY TO CARD No. 26.

In decimal operations, Long Division is preferable to Short Division, on account of knowing where to place the dividing point. This point is the only difficulty to be met with in decimals. It is a period called the separatrix, to separate decimals from integers, and integers from decimals.*

When we shall readily know where to place this point, decimals will become as familiar as integers; and greatly expedite business in frequent instances. To know how, and where to place this point, let us attend to a few particular rules.

DIVISION.

1. Count the decimal places in the dividend. Suppose them to be two as in lesson first.

2. Count the number of decimals in the divisor, if any, then begin at the right hand figure in the quo

*Integer signifies a whole number; as, a pound, an ounce, a bushel, gallon, &c. "The whole of any thing."

tient, and continue counting towards the left, till you have a number equal to the decimals in the dividend: there place the separatrix or point.

3. But if the number of figures in the quotient fall short, then prefix ciphers on the left till your counting is completed: there place the point.

4. When the dividend is less than the divisor, annex ciphers on the right according to discretion.

NOTE. In forming fractions into decimals, we consider Numerator, and Denominator, as two distinct whole numbers.

For instance, observe lesson first. We place 1 for a dividend, and 4 for a divisor. We then see that 4 cannot be contained in 1; therefore place a point and annex ciphers to the right of the point.

These ciphers show us how many decimal parts are contained in the dividend; and that the digit,* 1, is considered as an integer.

In the next place we proceed with Division; when done, begin to count and see where the point must be placed in the quotient. As there is no decimal place in the divisor of lesson 1, begin at the right hand of the quotient and count towards the left, till having a number equal to the decimal places in the dividend: then place our point as on the left side of .25 hundredths. This counting proves its being placed right at first.

KEY TO CARD No. 26.

LESSON 2,

What is the decimal of?

The numerator is 1. 2)1.0(.5|| In this lesson it is The denominator is 2. 10 necessary to annex

one cipher only; consequently there is but one decimal place in the dividend.

And as there is but one

figure in the quotient, the point must be placed on the left of that figure. Thus, .5 tenths.

*Digit, signifies any number under ten, as, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0*

NOTE.-The separatrix or point, is in the place of units when reading decimals, and instead of reading units, tens, hundreds, &c. towards the left as in whole numbers, we begin with the point, calling it units, and read to the right.

EXAMPLES.

Call the point, units. .25 units, tens, hundreds = twenty-five hundredths. .5 units, tens: five tenths. .075 units, tens, hundreds, thousands: seventy-five thousandths.

LESSON 3.

What is the decimal of ?What is the decimal of?? 4)3.00(.5 hundredths,

LESSON 5.

Answer, .875

8)7.000(.875

64

.60

56

.40

40

.100

100

LESSON 6.

Digression 1st.

What is the amount of .25 .5 .75 .875 and .95 of

Ans. .95 a pound?

Rule in Addition.

Begin at the period or units, and place tens under tens, hundreds under hundreds, &c. towards the right; then and as in whole numbers carrying at every ten.

.25

.5

.75

tenths.

hundredths. In words, Three pounds, land three hundred and twenty-five thousandths of a pound. Or, three tenths, two hundredths, and five thousandths of a pound.

hundredths. .875 thousandths. .95 hundredths.

3.325

How shall we know where

Answer, £3.325 to place the decimal point in Addition?

RULE.

Count that number which has the most decimal places: then begin at the right of the amount and count as many towards the left; there place the point. The figures on the left of the point are whole numbers. KEY TO CARD No. 26.

LESSON 7.

Digression 2nd.

How shall we find the value of £3.325 in pounds, shillings and pence?

RULE.

Multiply the decimals by the denominations in pounds, shillings, and pence; the figures that happen on the left of the point, are whole numbers.

What is the use of having several sums of money

formed into Decimals?